Characterizing vector fields

*Some* vector fields, $\myv F$, represent the gradient of a function, $\myv F=\myv \grad f$.

In the next few class sessions we'll be trying to figure out:

• How to tell if a vector field is the gradient of a function,
• How to find the function $f$, given $\myv F$.
• How this makes some line integrals *a lot* easier to evaluate.

Gradients

The gradient of a scalar field is a vector field. $$ \myv \grad f(x,y)=f_x(x,y)\uv i+f_y(x,y)\uv j=\myv v(x,y)$$

$$\myv \grad f(x,y,z)=f_x(x,y,z)\uv i+f_y(x,y,z)\uv j+f_z(x,y,z)\uv k=\myv v(x,y,z)$$

Gradient Fields

Some (but not all) vector fields are gradient fields. Consider $f(x,y)=x^2+xy+y^2$. The gradient is $$\myv \grad f=\langle f_x,f_y \rangle = (2x+y)\uv i+ (2y+x)\uv j.$$

Graphing the function $f$ below in contour plot. What can you say about

• The direction of the vectors in the vector field $\myv v=\myv \grad f$?
• Where are the vectors representing $\myv v$ longer or shorter?

In Mathematica

Plotting contours and gradient vector fields together... $$f(x,y)=\frac{x^2}{4}+\frac{y^2}{9}$$ Contours for $f=1,2,9$.

You'll try a couple of these in the lab...

• $f(x,y)=xy-2x$
• $f(x,y)=\sin(x)+\sin(y)$
• $f(x,y)=\sin(x+y)$

Conservative fields

Consider the vector field... $$\myv F(x,y)=2x\uv i+y\uv j$$

Can you find--or actually, just guess--a function $f(x,y)$ such that $$\myv F(x,y) = \myv \grad f(x,y)?$$

If such a function $f$ exists:

• $f(x,y)$ is called a potential function for the vector field $\myv F$.
• $\myv F(x,y)$ is a conservative vector field.