2) Curvilinear coordinate systems and vector calculus

Question 1

What is the Magnitude of a vector

Answer 1

Question 2

What is the Dot Product of two Vectors

Answer 2

Question 3

What is the Cross Product of two Vectors

Answer 3

Question 4

What are the key properties and implications of dot products and cross products between vectors

Answer 4

• Orthogonal vectors have dot product 0.
• Parallel unit vectors have dot product 1
• u × v = 0 if u and v have the same direction, i.e. if u = αv for some constant α

Question 5

What is the Grad(ient) Operator (∇)

Answer 5

Question 6

What are the properties of the grad operator

Answer 6

• Has direction perpendicular to the level surfaces
• Points in the direction of increasing u
• Has magnitude equal to the rate of change of u in this direction

Question 7

What is the Divergence of a Vector Field

Answer 7

Question 8

What is the Curl of a Vector Field

Answer 8

Question 9

What is the Laplacian operator

Answer 9

Question 10

What are the inputs and outputs of the four main operators

Answer 10

• grad: scalar → vector
• div: vector → scalar
• curl: 3D vector → 3D vector
• Laplacian: scalar → scalar

Question 11

What makes the nabla (∇) operator universal in vector calculus

Answer 11

• It operates independently of the coordinate system, allowing equations to be formulated generally and applied in any system
• ∇x(∇f)=0 ∇⋅(∇×f)=0 hold true in all coordinate systems, demonstrating its universal applicability.

Question 12

What is the alternative definition of the divergence

Answer 12

Question 13

What is the alternative definition of the curl

Answer 13

Question 14

What are curvilinear coordinates and how are they parametrised

Answer 14

• Curvilinear coordinates are a set of coordinates where the system’s grid lines may be curved, allowing flexibility in describing locations in space
• They are denoted by v1, v2, v3

Question 15

How does the position vector change with small changes in curvilinear coordinates, and what are the corresponding basis vectors

Answer 15

Question 16

When is a curvilinear coordinate system orthogonal

Answer 16

Question 17

When is a curvilinear coordinate system right-handed

Answer 17

If e1 × e2 = e3

Question 18

What are Cylindrical Polar Coordinates

Answer 18

Question 19

What are the tangent vectors and scale factors of the Cylindrical Polar Coordinates

Answer 19

Question 20

What are the basis vectors of the Cylindrical Polar Coordinates

Answer 20

Question 21

What are the expressions for differential length, surface, and volume elements in cylindrical polar coordinates

Answer 21

Question 22

What are Spherical Polar Coordinates

Answer 22

Question 23

What are the basis vectors of the Spherical Polar Coordinates

Answer 23

Question 24

What are the expressions for differential length, surface, and volume elements in Spherical Polar Coordinates

Answer 24

Question 25

What are Parabolic coordinates

Answer 25

Question 26

What are the basis vectors of the Parabolic coordinates

Answer 26

Question 27

How is the gradient operator (∇) expressed in curvilinear coordinates

Answer 27

Question 28

How is the divergence of a vector field expressed in curvilinear coordinates

Answer 28

Question 29

How is the curl of a vector field expressed in curvilinear coordinates

Answer 29

Question 30

How is the Laplacian expressed in orthogonal curvilinear coordinates

Answer 30