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Chapter 14: Problem 14

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

Short Answer

Expert verified
The gradient field for the given function \(f(x, y)=x^{2}-y^{2}\) is \(\nabla f= (2x, -2y)\). It can be graphically represented using a tool like Wolfram Alpha, Mathematica, or any other CAS software.

Step by step solution

01

Find the Partial Derivative w.r.t. to x

To find the partial derivative of the function \(f(x, y)=x^{2}-y^{2}\) with respect to \(x ( \frac{\partial f}{\partial x})\), we treat \(y\) as a constant. Derivation of \(x^{2}\) gives us \(2x\), thus, \(\frac{\partial f}{\partial x}= 2x\).
02

Find the Partial Derivative w.r.t. to y

Similarly, to find the partial derivative of the function \(f(x, y)=x^{2}-y^{2}\) with respect to \(y ( \frac{\partial f}{\partial y})\), we consider \(x\) to be constant. Derivation of \(-y^{2}\) gives us \(-2y\), thus, \(\frac{\partial f}{\partial y}= -2y\).
03

Compute the gradient field

The gradient of the function is the vector \(\nabla f= (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\) . Substituting our earlier findings, we obtain the gradient field as \(\nabla f= (2x, -2y)\).
04

Graph the gradient field using a CAS

At this step, we will move the solution to any computer algebra system (like Wolfram Alpha, Mathematica, Maple etc.) to graph the gradient field. Each vector of the gradient field on the plane is plotted at its corresponding point. This step is beyond the text explanation and done practically using software.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are essential in multivariable calculus. If you have a function with more than one variable, a partial derivative enables you to see how a specific variable in the function affects its value, while keeping all other variables constant. In this context, when we computed the partial derivative of the function \(f(x, y) = x^{2} - y^{2}\) with respect to \(x\), we treated \(y\) as a constant.
  • The partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x} = 2x\).
  • Similarly, the partial derivative with respect to \(y\) is \(\frac{\partial f}{\partial y} = -2y\).

By understanding these derivatives, we can form a better picture of what the gradient field of a function looks like, essentially giving us a map of how the function changes in different directions in space.
Computer Algebra System
A computer algebra system (CAS) is a software tool designed to solve mathematical problems just like a human would, but with greater speed and accuracy. CAS tools are widely used in mathematics to perform symbolic calculations like differentiation, integration, and plotting functions.
Using a CAS to graph the gradient field provides a visual understanding of the function. After computing the gradient \(abla f = (2x, -2y)\), a CAS can plot this vector field to show how the function changes at different points. This graphical approach gives a tangible insight into the behaviour patterns of functions involving multiple variables.
  • Popular CAS tools include Wolfram Alpha, Mathematica, and Maple.
  • They allow you to not just perform mathematical operations, but also visualize complex mathematical ideas.
Vector Calculus
Vector calculus is a branch of mathematics that deals with vector fields and operations on these fields. It's particularly important in physics and engineering because many physical systems are best described by vectors.
In the exercise, the gradient field \(abla f = (2x, -2y)\) is a simple example of how vector calculus is applied. Here, the gradient is represented as a vector field showing the direction and rate of the fastest increase of the function \(f(x, y) = x^2 - y^2\).
  • Every point in the field has a vector, showing the steepest path uphill in the function's landscape.
  • Understanding such changes is crucial when analyzing things like fluid flow or electromagnetic fields.
Vector calculus uses tools like divergence, curl, and gradient to model and analyze these behaviors efficiently.

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Most popular questions from this chapter

Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of \(z=\sqrt{4-x^{2}-y^{2}}\) above the \(x y\) -plane with \(\mathbf{n}\) upward, \(\mathbf{F}=\left\langle z x^{2}, z e^{x y^{2}}-x, x \ln y^{2}\right\rangle\)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the intersection of \(z=x^{2}+y^{2}-4\) and \(z=y-1\) oriented clockwise as viewed from above, \(\mathbf{F}=\left\langle\sin x^{2}, y^{3}, z \ln z-x\right\rangle\)

Find the mass and center of mass of the region. The portion of the plane \(x+2 y+z=4\) above the region bounded by \(y=x^{2}\) and \(y=1, \rho(x, y, z)=y\)

Find equations for the flow lines. $$\left\langle 2 y, 3 x^{2}\right\rangle$$

Find the flux of \(\langle x, y, 0\rangle\) across the portion of \(z=c \sqrt{x^{2}+y^{2}}\) below \(z=1 .\) Explain in physical terms why this answer makes sense.
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