Fields and Fluids

Question 1

Volume integral in spherical coordinates

Answer 1

Question 2

Total charge of a configuration

Answer 2

Question 3

Electric field of a configuration

Answer 3

Choose a Gaussian surface
Then use Gauss’ law in integral form:

Question 4

Coulomb’s law

Answer 4

Question 5

Area of a sphere

Answer 5

4𝜋𝑟^2

Question 6

Finding components of curl and divergence

Answer 6

• Divergence (.) is already its own component
• Curl (x) need to take its i-th component

Question 7

Properties of Tensors

Answer 7

Question 8

Decomposition of rank-2 tensors

Answer 8

Question 9

Decomposition of a symmetric tensor

Answer 9

Question 10

Transformation rule of a rank-2 tensor

Answer 10

Question 11

Tensor components

Answer 11

Question 12

Symmetry of 𝜀 and 𝛿

Answer 12

𝜀 - anti-symmetric
𝛿 - symmetric

Question 13

Vorticity

Answer 13

𝜔 = ∇ × 𝑞

Question 14

Circulation

and the circulation around a circle, centre origin, radius r

Answer 14

Question 15

Stream functions

Answer 15

Question 16

Constant streamline equations

Answer 16

Question 17

Complex velocity

Answer 17

Question 18

Wave equations in a vacuum

Answer 18

Question 19

Plane wave equations

Answer 19

Question 20

Electrostatics equations

Answer 20

Question 21

Magnetostatics equations

Answer 21

Question 22

Gauss’ law in integral form

Answer 22

Question 23

Ampère’s law in integral form

Answer 23

Question 24

Electromotive force

Answer 24

Question 25

Electric flux through a sphere

Answer 25

Question 26

Rotation matrix, about the z-axis

Answer 26

Question 27

Reflection matrix in a line L through the origin

Answer 27

Question 28

Stagnation points of a system

Answer 28

Points of vanishing velocity, where:
for q(u,v), u = v = 0

Question 29

Electric flux of a point charge

Answer 29

Question 30

Polar coordinates

Answer 30

r^2 = x^2 + y^2
x = r cos(theta)
y = r sin(theta)

Question 31

Matrix product in index notation

Answer 31

Question 32

Acceleration

Answer 32

Question 33

Compressibility

Answer 33

A fluid is incompressible if the rate of change of its density is zero: D𝜌/Dt