**Maintainer:**admin

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

- 1 Find the flux of a field given its surface
- 2 Prove an identity involving the del operator
- 3 Surface integrals of a scalar function
- 4 Computing the curl of a vector field
- 5 Potential functions of vector fields
- 6 Find a vector function given its curl
- 7 Line integrals - vector fields
- 8 Line integrals - cartesian coordinates

*1*Find 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)

*1.1*General solution¶

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

*1.2*Examples¶

- Assignment 1, questions 5, 6, and 3 (c)
- Winter 2010 final, question 3

*2*Prove an identity involving the del operator¶

Won't be necessary

*2.1*Examples¶

- Assignment 1, question 7
- Assignment 4, question 2

*3*Surface integrals of a scalar function¶

Meh

*3.1*Examples¶

- Assignment 2, question 2

*4*Computing the curl of a vector field¶

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

*4.1*Examples¶

- Assignment 2, questions 3 (b) and 4 (a)
- Winter 2010 final, question 4 (a)

*5*Potential 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?)

*5.1*Examples¶

- Assignment 2, question 4 (b)
- Winter 2010 final, question 4 (b)
- Winter 2011 final, question 5

*6*Find a vector function given its curl¶

*6.1*Examples¶

- Assignment 4, question 1

*7*Line 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.

*7.1*General 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.

*7.2*Examples¶

- Assignment 2, questions 3 (a) and 4 (c)
- Winter 2006 final, question 6

*8*Line integrals - cartesian coordinates¶

Same as above but Cartesian coordinates instead of a parametrised curve

*8.1*General 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.

*8.2*Examples¶

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