MATH 251 Lecture 27

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Vector Fields

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

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

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

... diffy Q's ... :)

${\vec {F}}$ is called "conservative" if ${\vec {F}}=\nabla f$ 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 $f_{y}=-x+C'(y)=-x+0$

Another Antidifferentiation Example

${\begin{cases}f_{x}=-y\\f_{y}=x\end{cases}}$

$f=-xy+C(y)$

Based on our function above, $f_{x}=-x+C'(y)=x$, so $C'(y)=2x$

This is unsolvable.