Complete Theory

Worked Examples

Example 1Conservative field: find the potential

Example 2Green's theorem

Example 3Divergence theorem

Example 4Stokes' theorem

Example 5Vortex field: irrotational but not conservative

Exercise 1Div and curlMedium

📋 Problem to solve

Compute div and curl of

\mathbf{F}=(xy,yz,zx)

Exercise 2PotentialMedium

📋 Problem to solve

Find

U

for

\mathbf{F}=(2x,2y)

Exercise 3GreenHard

📋 Problem to solve

Area of the disk

r\leq a

via Green's theorem.

Exercise 4DivergenceHard

📋 Problem to solve

Flux of

\mathbf{F}=(x^2,y^2,z^2)

over the cube

[0,1]^3

Exercise 5StokesHard

📋 Problem to solve

Via Stokes:

\oint_\gamma(-y,x,0)\cdot d\mathbf{r}

\gamma

: circle

r=1

z=0

Exercise 6IdentitiesMedium

📋 Problem to solve

Verify

\nabla\cdot(\nabla\times\mathbf{F})=0

for

\mathbf{F}=(yz,zx,xy)

📚

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