The Temperature Distribution Function

Oct 13, 2025 by windailys.com 111 views

Introduction

In this article, we will explore the temperature distribution function given by $T(x, y, z) = 300 e^{-x^2 - 5y^2 - 9z^2}$ , where $T$ is measured in ${ }^{\circ} C$ and $x, y, z$ in meters. We will first analyze the function and then find the rate of change of the temperature at a point $(x, y, z)$ .

The Temperature Distribution Function

The temperature distribution function is given by $T(x, y, z) = 300 e^{-x^2 - 5y^2 - 9z^2}$ . This function represents the temperature at a point $(x, y, z)$ in a three-dimensional space. The function is a product of two terms: the constant term $300$ and the exponential term $e^{-x^2 - 5y^2 - 9z^2}$ .

Properties of the Exponential Function

The exponential function $e^{-x^2 - 5y^2 - 9z^2}$ has several important properties that we need to understand in order to analyze the temperature distribution function. The exponential function is a continuous and differentiable function, and it has a maximum value of $1$ at $x = y = z = 0$ . The function is also symmetric about the origin, meaning that $f(-x, -y, -z) = f(x, y, z)$ for all $x, y, z$ .

The Rate of Change of the Temperature

The rate of change of the temperature at a point $(x, y, z)$ is given by the partial derivatives of the temperature distribution function with respect to $x, y, z$ . We will first find the partial derivatives of the function and then use them to find the rate of change of the temperature.

Partial Derivatives of the Temperature Distribution Function

To find the partial derivatives of the temperature distribution function, we will use the chain rule and the product rule of differentiation.

Partial Derivative with Respect to $x$

The partial derivative of the temperature distribution function with respect to $x$ is given by:

\frac{\partial T}{\partial x} = 300 e^{-x^2 - 5y^2 - 9z^2} \cdot (-2x)

Simplifying the expression, we get:

\frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2}

Partial Derivative with Respect to $y$

The partial derivative of the temperature distribution function with respect to $y$ is given by:

\frac{\partial T}{\partial y} = 300 e^{-x^2 - 5y^2 - 9z^2} \cdot (-10y)

Simplifying the expression, we get:

\frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2}

Partial Derivative with Respect to $z$

The partial derivative of the temperature distribution function with respect to $z$ is given by:

\frac{\partial T}{\partial z} = 300 e^{-x^2 - 5y^2 - 9z^2} \cdot (-18z)

Simplifying the expression, we get:

\frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2}

The Rate of Change of the Temperature

The rate of change of the temperature at a point $(x, y, z)$ is given by the partial derivatives of the temperature distribution function with respect to $x, y, z$ . We can use the partial derivatives to find the rate of change of the temperature in any direction.

For example, if we want to find the rate of change of the temperature in the $x$ -direction, we can use the partial derivative with respect to $x$ :

\frac{dT}{dx} = \frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2}

Similarly, if we want to find the rate of change of the temperature in the $y$ -direction, we can use the partial derivative with respect to $y$ :

\frac{dT}{dy} = \frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2}

And if we want to find the rate of change of the temperature in the $z$ -direction, we can use the partial derivative with respect to $z$ :

\frac{dT}{dz} = \frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2}

Conclusion

In this article, we have analyzed the temperature distribution function given by $T(x, y, z) = 300 e^{-x^2 - 5y^2 - 9z^2}$ and found the rate of change of the temperature at a point $(x, y, z)$ . We have used the partial derivatives of the function to find the rate of change of the temperature in any direction. The results can be used to understand the behavior of the temperature distribution function and to make predictions about the temperature at different points in space.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Multivariable Calculus, 6th edition, James Stewart

Appendix

A.1 Partial Derivatives of the Temperature Distribution Function

The partial derivatives of the temperature distribution function are given by:

\frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2}

\frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2}

\frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2}

A.2 The Rate of Change of the Temperature

For example, if we want to find the rate of change of the temperature in the $x$ -direction, we can use the partial derivative with respect to $x$ :

\frac{dT}{dx} = \frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2}

Similarly, if we want to find the rate of change of the temperature in the $y$ -direction, we can use the partial derivative with respect to $y$ :

\frac{dT}{dy} = \frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2}

And if we want to find the rate of change of the temperature in the $z$ -direction, we can use the partial derivative with respect to $z$ :

\frac{dT}{dz} = \frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2}$$<br/> **The Temperature Distribution Function and Its Rate of Change: Q&A** ===========================================================

Introduction

In our previous article, we analyzed the temperature distribution function given by $T(x, y, z) = 300 e^{-x^2 - 5y^2 - 9z^2}$ and found the rate of change of the temperature at a point $(x, y, z)$ . In this article, we will answer some frequently asked questions about the temperature distribution function and its rate of change.

Q: What is the temperature distribution function?

A: The temperature distribution function is a mathematical function that represents the temperature at a point $(x, y, z)$ in a three-dimensional space. It is given by $T(x, y, z) = 300 e^{-x^2 - 5y^2 - 9z^2}$ .

Q: What is the rate of change of the temperature?

A: The rate of change of the temperature at a point $(x, y, z)$ is given by the partial derivatives of the temperature distribution function with respect to $x, y, z$ . We can use the partial derivatives to find the rate of change of the temperature in any direction.

Q: How do I find the rate of change of the temperature in the $x$ -direction?

A: To find the rate of change of the temperature in the $x$ -direction, you can use the partial derivative with respect to $x$ :

\frac{dT}{dx} = \frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2} </span></p> <h2><strong>Q: How do I find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>-direction?</strong></h2> <p>A: To find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>-direction, you can use the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>3000</mn><mi>y</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{dT}{dy} = \frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span 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class="strut" style="height:1.2314em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">3000</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: How do I find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>-direction?</strong></h2> <p>A: To find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>-direction, you can use the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>5400</mn><mi>z</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{dT}{dz} = \frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1202em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">5400</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: What is the physical significance of the rate of change of the temperature?</strong></h2> <p>A: The rate of change of the temperature represents the rate at which the temperature changes in a particular direction. It can be used to understand the behavior of the temperature distribution function and to make predictions about the temperature at different points in space.</p> <h2><strong>Q: Can I use the rate of change of the temperature to find the temperature at a point?</strong></h2> <p>A: Yes, you can use the rate of change of the temperature to find the temperature at a point. By integrating the rate of change of the temperature with respect to the direction of interest, you can find the temperature at a point.</p> <h2><strong>Q: What are some common applications of the temperature distribution function?</strong></h2> <p>A: The temperature distribution function has many common applications in physics, engineering, and other fields. Some examples include:</p> <ul> <li>Modeling the temperature distribution in a heat transfer problem</li> <li>Analyzing the behavior of a thermodynamic system</li> <li>Designing a temperature control system</li> <li>Predicting the temperature at different points in a three-dimensional space</li> </ul> <h2><strong>Conclusion</strong></h2> <p>In this article, we have answered some frequently asked questions about the temperature distribution function and its rate of change. We hope that this article has been helpful in understanding the temperature distribution function and its rate of change.</p> <h2><strong>References</strong></h2> <ul> <li>[1] Calculus, 3rd edition, Michael Spivak</li> <li>[2] Multivariable Calculus, 6th edition, James Stewart</li> <li>[3] Thermodynamics, 2nd edition, C. J. Adkins</li> </ul> <h2><strong>Appendix</strong></h2> <h3>A.1 Partial Derivatives of the Temperature Distribution Function</h3> <p>The partial derivatives of the temperature distribution function are given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>600</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1202em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">600</span><span class="mord mathnormal">x</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>3000</mn><mi>y</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2314em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">3000</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>5400</mn><mi>z</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1202em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">5400</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <h3>A.2 The Rate of Change of the Temperature</h3> <p>The rate of change of the temperature at a point <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y, z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> is given by the partial derivatives of the temperature distribution function with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x, y, z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>. We can use the partial derivatives to find the rate of change of the temperature in any direction.</p> <p>For example, if we want to find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>-direction, we can use the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>600</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{dT}{dx} = \frac{\partial T}{\partial x} = -600x e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1202em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">600</span><span class="mord mathnormal">x</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p>Similarly, if we want to find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>-direction, we can use the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>3000</mn><mi>y</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{dT}{dy} = \frac{\partial T}{\partial y} = -3000y e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2314em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">3000</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p> <p>And if we want to find the rate of change of the temperature in the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>-direction, we can use the partial derivative with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>T</mi></mrow><mrow><mi>d</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>T</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>5400</mn><mi>z</mi><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{dT}{dz} = \frac{\partial T}{\partial z} = -5400z e^{-x^2 - 5y^2 - 9z^2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1202em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">5400</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">5</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">9</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>