HTSEFP: Vector calculus

Vector calculus. Includes some stuff on flux and surface integrals, but nothing on Green's, Stokes', or the divergence theorem.

1Find the flux of a field given its surface¶

Given the formula for the vector field and the formula for the surface (2- or 3-dimensional)

1.1General solution¶

Take the cross-product of the partial derivs, dot it with $F(r(u,v))$, integrate over limits of region

2Prove an identity involving the del operator¶

Won't be necessary

2.1Examples¶

• Assignment 1, question 7
• Assignment 4, question 2

3Surface integrals of a scalar function¶

Meh

3.1Examples¶

• Assignment 2, question 2

4Computing the curl of a vector field¶

Easy, take it's curl (see formulas page), if it's 0 then it's conservative

5Potential functions of vector fields¶

And using that to compute a line integral (given that the vector field is conservative - wait, is that just a given if you can find the potential function?)

6Find a vector function given its curl¶

6.1Examples¶

• Assignment 4, question 1

7Line integrals - vector fields¶

This is the one where you have $\displaystyle \int_C \vec F \cdot \,d\vec r$, where $\vec F$ is some vector field (given) and $C$ is some boundary curve (could be given in terms of its formula or a description of its shape or whatever) and no mention is made of Green's theorem.

7.1General solution¶

Parametrise the curve $C$, resulting in $r(t)$ (possibly more than one, if you need to parametrise in segments). Then, use the following formula:

$$\int_a^b \vec F(r(t)) \cdot r'(t)\,dt$$

where $a$ and $b$ are the lower and upper limits for $t$ along the curve, respectively.

8Line integrals - cartesian coordinates¶

Same as above but Cartesian coordinates instead of a parametrised curve

8.1General solution¶

Parametrise the curve $C$ and use the x and y components of $r(t)$ to transform x and y into functions of t. Then integrate with respect to t, using the limits for $t$ from the curve you parametrised.

8.2Examples¶

• I'm sure there are some though I can't find any at the moment