MATH 251 Lecture 27

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Vector Fields

A book on vector Calculus: Div, Grad, Curl, and all that

$\overrightarrow{F}(x,y)=a(x,y)\hat{ı}+b(x,y)\hat{ȷ}$ gives a vector at every point $(x,y)$. Similarly, $\overrightarrow{F}(x,y,z)=a(x,y,z)\hat{ı}+b(x,y,z)\hat{ȷ}+k(x,y,z)\hat{k}$ gives a vector at every point $(x,y,z)$

The vectors are understood to originate at the point for which they are plugged into $\overrightarrow{F}$. Imagine oodles of arrows! Also think of it as the velocity field of a fluid.

... diffy Q's ... :)

$\overrightarrow{F}$ is called "conservative" if $\overrightarrow{F}=\mathrm{\nabla }f$ for some scalar function $f$. This function $f$ is also called the "potential"

Example based on definition

Find $f(x,y)$ given $f_x=3x^2-y$ and $f_y=-x$

Antidifferentiation: From the first function: $\int 3x^2-y\mathrm{d}x=x^3-xy+C(y)$

Notice how the constant can be any function with respect to $y$ (i.e. not w/r/t $x$)

From the second function: $C(y)$ must be a constant because $f_y=-x+C^{\prime }(y)=-x+0$

Another Antidifferentiation Example

$\{\begin{array}{l}{f}_x=-y\\ f_y=x\end{array}$

$f=-xy+C(y)$

Based on our function above, $f_x=-x+C^{\prime }(y)=x$, so $C^{\prime }(y)=2x$

This is unsolvable.