Multidimensional integration

Question 1

Describe the transformation between cartesian coordinates (x,y,z) and spherical coordinates

Answer 1

x = rsinθcosϕ

y = rsinθsinϕ

z = rcosθ

Question 2

Describe the transformation between cartesian coordinates (x,y,z) and cylindrical coordinates

Answer 2

x = pcosϕ

y = psinϕ

z = z

Note that cylindrical coordinates are just polar coordinates in 3D with z for the third variable

Question 3

Give the equation of a line integral for the vector field G(r)

Answer 3

Note that the value of the line integral along a path C is independent of the parametrisation of the path

Question 4

Give the equation of a loop integral for vector field G(r)

Answer 4

Question 5

Give the equation of an area integral for any scalar field f(r)

Answer 5

Question 6

Give the transformation between cartesian coordinates and polar coordinates

Answer 6

x = rcosφ

y = rsinφ

Question 7

Give the line element in polar coordinates

Answer 7

Question 8

Give the area element in polar coordinates

Answer 8

Question 9

Give the volume element for the function f(r) in cartesian coordinates

Answer 9

Question 10

Give the line element in cylindrical coordinates

Answer 10

Question 11

Give the volume element in cylindrical coordinates

Answer 11

Question 12

Give the line element in spherical coordinates

Answer 12

Question 13

Give the volume element in spherical coordinates

Answer 13

Question 14

Give the general equation for the surface integral for any scalar field

f(r) = f(x,y,z)

Answer 14

Question 15

Give the general equation for the total flux of a vector field G(r)

Answer 15

Note that total flux is given by the surface area, as only perpendicular components of G contribute to flux

Question 16

Define dS

Answer 16

S is parametrised by r = r(s,t)

Note that x denotes the cross product

Question 17

Give the relationship between dV and dS

Answer 17

dV = dSdr

Question 18

State Gauss’s theorem

Answer 18

Fhe flux of a vector field G(r) through a closed surface S = ∂V is equal to the integral of the divergence of G over the enclosed volume V

Question 19

State Stokes’s theorem

Answer 19

The loop integral of a vector field G(r) around the boundary C = ∂S of an open surface S is equal to the flux of the curl of the vector field

Question 20

Define flux

Answer 20

Flux describes an effect that appears to move through a surface or substance

Question 21

Define dS when z = g(x,y)

Answer 21

Question 22

Define the vector product or cross product for two vectors a and b

Answer 22