Problem 13 - Calculus: Early Transcendental Functions

Question

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=x^{2}+y^{2}$$

Step-by-Step Solution

Answer

The gradient field corresponding to the function \(f(x, y) = x^{2} + y^{2}\) is \(\nabla f = (2x, 2y)\). It can be graphed using a Computer Algebra System as a vector field where each vector at a point in the plane represents the direction and magnitude of the maximum rate of increase of the function at that point.

Step 1: Calculating the Gradient of \(f\)

The gradient of a function is a vector field. It is computed by finding the partial derivative of the function with respect to each variable, \(x\) and \(y\) in this case. It is denoted as \(\nabla f\) or \(grad f\). If \(f(x, y) = x^{2} + y^{2}\), then the gradient of \(f\) is \(\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\). Calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) separately. The partial derivative of \(f\) with respect to \(x\) is \(\frac{\partial f}{\partial x} = 2x\) and with respect to \(y\) is \(\frac{\partial f}{\partial y} = 2y\).

Step 2: Forming the Gradient Field

After finding the partial derivatives, combine them i.e. \(\nabla f = (2x, 2y)\). This pair of functions represents the gradient field of \(f\).

Step 3: Graphing the Gradient Field using a CAS

The gradient field is graphically represented as a vector field. In a vector field, at each point in the plane, we draw a vector that represents the components of the function at that point. A CAS (Computer Algebra System) can be used to graph the vector field derived from \(\nabla f\). Each vector in the field shows the direction and the magnitude of the maximum rate of increase of the function at a particular point.

Key Concepts

Partial Derivative

Understanding partial derivatives is essential when dealing with functions of multiple variables, like in the expression \(f(x, y) = x^{2} + y^{2}\). A partial derivative measures how a function changes as only one of its input variables changes, while keeping the other variables constant. This concept is crucial because it allows us to determine how each variable independently influences the function.

To find a partial derivative, you differentiate the function with respect to the variable of interest, treating all other variables as constants. In the given exercise:

• The partial derivative of \(f\) with respect to \(x\) is found by differentiating \(f\) while treating \(y\) as a constant. This yields \(\frac{\partial f}{\partial x} = 2x\).


• Similarly, the partial derivative with respect to \(y\) is computed by treating \(x\) as constant, resulting in \(\frac{\partial f}{\partial y} = 2y\).

These two partial derivatives are then used to form the gradient vector, which provides important insights into the behavior of the function in a multivariable context.

Vector Field

A vector field is a visual representation that shows how vectors are assigned to each point in a space. In our case, the gradient field is a specific type of vector field generated from the function \(f(x, y) = x^{2} + y^{2}\).

The gradient \(abla f\), calculated as \((2x, 2y)\), gives us the direction and magnitude of the greatest increase of the function at any point \((x, y)\). This means that at any point on the plane, the vector provides information about which direction to move to increase the function's value most rapidly.

Visualizing this vector field involves drawing vectors at various points on the plane. Each vector's length represents the magnitude of the gradient at that point, while its direction shows the way in which the function is increasing. By looking at the vector field, one can immediately see how the function behaves across different regions.

Computer Algebra System (CAS)

A Computer Algebra System, conveniently referred to as a CAS, is a software tool that aids in performing symbolic mathematics efficiently. It can be incredibly helpful for plotting complex functions or vector fields, such as the gradient field of \(f(x,y)=x^2+y^2\).

Using a CAS to graph a vector field offers several advantages. It eliminates the tedious manual calculations and drawings, providing precise and comprehensive visualizations. This ensures a clear comprehension of the function's behavior throughout its domain.

In our exercise, utilizing a CAS helps in accurately plotting the gradient field \(abla f = (2x, 2y)\). The software can quickly create a visual representation, displaying vectors at various points that reveal both the direction and strength of the function's increase at each location. This visual aid can be immensely beneficial for analyzing complex mathematical functions and interpreting their impacts in a given context.