Question

The temperature at the point \((x, y)\) on a metal plate is modeled by \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}, x \geq 0, y \geq 0\) (a) Use a computer algebra system to graph the temperature distribution function. (b) Find the directions of no change in heat on the plate from the point (3,5) . (c) Find the direction of greatest increase in heat from the point (3,5)

Short answer

Using a computer algebra system, the function graph can be obtained. The directions of no change in heat are obtained by setting the first order partial derivatives to zero. Lastly, the direction of greatest increase corresponds to the direction of the gradient vector of the function evaluated at the point (3,5).

Step-by-step solution

Graph of the Function

To visualize the temperature distribution, a graph should be plotted using a computer algebra system like MATLAB, Python or Mathematica. The function \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}\), for \(x \geq 0, y \geq 0\), should be entered into the system and graphed.

Finding Direction of No Change in Heat

At directions of no change, the partial derivatives of the function with respect to x and y should be zero. So, find the first order derivatives, set them to zero, and solve to find such directions. Partial derivative formulas are given by: \[\frac{\partial T}{\partial x}=400 e^{-\left(x^{2}+y\right) / 2} \cdot -x\]and\[\frac{\partial T}{\partial y}=400 e^{-\left(x^{2}+y\right) / 2} \cdot -1 \]Set these to zero and solve for x and y respectively.

Finding Direction of Greatest Increase in Heat

The direction of maximum increase in temperature corresponds to the direction of the gradient vector of the function, which is the vector of the first order partial derivatives. It's calculated as follows:\[\nabla T = \left< \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}\right> \]We evaluate this at the point (3,5).

Key concepts

Computer Algebra System Graph

When studying temperature distribution on a metal plate, visualization plays a key role in understanding the concept. To facilitate this, computer algebra systems (CAS) such as MATLAB, Python with libraries like Matplotlib, or Wolfram Mathematica are extremely useful. They enable us to input a temperature distribution function like \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}\), and produce a three-dimensional surface graph.

For beginners, simply input the function into the CAS, specifying the domain for \(x \geq 0\) and \(y \geq 0\). The computer will then calculate the temperature at numerous points across the specified domain and plot these points in three-dimensional space, providing a visual representation of how temperature changes across the surface of the metal plate. This graph helps in grasping the concept of temperature gradients and distribution over the plane.

Partial Derivatives

Partial derivatives are a cornerstone of multivariable calculus, especially in problems involving temperature distribution. They allow us to analyse how temperature changes in one direction independently of the other. In mathematical terms, the partial derivative of a function \(f(x, y)\) with respect to x, denoted as \(\frac{\partial f}{\partial x}\), describes how \(f\) changes as \(x\) varies, holding \(y\) constant, and vice versa.

To find the directions of no change in heat on the metal plate from the point (3,5), you need to set the first order partial derivatives of the temperature function to zero. This is because at these points, the temperature doesn’t increase or decrease as you move slightly in any direction. The equations for these derivatives are already provided in the step-by-step solution, and they need to be solved to determine the points where there's no temperature change.

Gradient Vector

The gradient vector is a concept that essentially communicates the direction and rate of fastest increase of a function. Represented as \(abla f\) for a function \(f\), it is a vector composed of all of its first order partial derivatives. In the context of temperature, the gradient of the function \(T(x, y)\) points in the direction of the greatest temperature increase.

For a function of two variables, T(x, y), the gradient vector is \(abla T = \left\langle \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}\right\rangle\). When you evaluate this at a specific point, like (3,5), the resulting vector points towards the direction where the temperature increases most sharply from that point. Understanding the gradient vector is crucial in fields such as thermodynamics, where it’s important to know which way heat will flow.