Vector Analysis 2

Question 1

single integral ∫ V, where V = xyz

Answer 1

= ∫ V dr
= ∫ ( x(u) * y(u) * z(u) ) ( dx/du du î + dy/du du ĵ + dz/du du k̂ )
= […] î + […] ĵ + […] k̂

where x = x(u)
y = y(u)
z = z(u)

Question 2

double integral ∫∫ f(x, y) dxdy

Answer 2

= ∫∫ f(u, v) (jacobian) du dv

Question 3

jacobian

Answer 3

|δ (x, y) | = δx/δu δy/δv - δx/δv δy/δu
|———| .
|δ (u, v) | .

Question 4

What does line intergral tell us

Answer 4

the work done in moving a particle from point a to b

Question 5

determine the volume of a vector

Answer 5

= ∫ ∫ ∫ xî + yĵ + zk̂ dx dy dz

Question 6

determine the surface integral

Answer 6

= ∫ V dS ň
- determine φ, = x+ y + z
- calculate gradient ň = Δφ / |Δφ|, = ( î d/dx, ĵ d/dy, k̂ d/dz) / sqrt( d/dx ^2 + d/dy ^2 + d/dz ^2)
- dS = r dΘ dz