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Flashcards in Multidimensional integration Deck (22)
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1

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

x = rsinθcosϕ

y = rsinθsinϕ

z = rcosθ

2

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

x = pcosϕ

y = psinϕ

z = z

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

3

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

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

4

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

5

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

6

Give the transformation between cartesian coordinates and polar coordinates

x = rcosφ

y = rsinφ

7

Give the line element in polar coordinates 

8

Give the area element in polar coordinates 

9

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

10

Give the line element in cylindrical coordinates

11

Give the volume element in cylindrical coordinates

12

Give the line element in spherical coordinates 

13

Give the volume element in spherical coordinates 

14

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

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

15

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

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

16

Define dS

S is parametrised by r = r(s,t) 

Note that x denotes the cross product 

17

Give the relationship between dV and dS

dV = dSdr

18

State Gauss's theorem 

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 

19

State Stokes's theorem 

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

20

Define flux

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

21

Define dS when z = g(x,y)

22

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