2) Curvilinear coordinate systems and vector calculus Flashcards
What is the Magnitude of a vector
What is the Dot Product of two Vectors
What is the Cross Product of two Vectors
What are the key properties and implications of dot products and cross products between vectors
- 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
What is the Grad(ient) Operator (∇)
What are the properties of the grad operator
- 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
What is the Divergence of a Vector Field
What is the Curl of a Vector Field
What is the Laplacian operator
What are the inputs and outputs of the four main operators
- grad: scalar → vector
- div: vector → scalar
- curl: 3D vector → 3D vector
- Laplacian: scalar → scalar
What makes the nabla (∇) operator universal in vector calculus
- 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.
What is the alternative definition of the divergence
What is the alternative definition of the curl
What are curvilinear coordinates and how are they parametrised
- 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
How does the position vector change with small changes in curvilinear coordinates, and what are the corresponding basis vectors
When is a curvilinear coordinate system orthogonal
What are Cylindrical Polar Coordinates
What are the tangent vectors and scale factors of the Cylindrical Polar Coordinates
What are the basis vectors of the Cylindrical Polar Coordinates
What are the expressions for differential length, surface, and volume elements in cylindrical polar coordinates
What are Spherical Polar Coordinates
What are the basis vectors of the Spherical Polar Coordinates
What are the expressions for differential length, surface, and volume elements in Spherical Polar Coordinates
What are Parabolic coordinates
What are the basis vectors of the Parabolic coordinates
How is the gradient operator (∇) expressed in curvilinear coordinates
How is the divergence of a vector field expressed in curvilinear coordinates
How is the curl of a vector field expressed in curvilinear coordinates
How is the Laplacian expressed in orthogonal curvilinear coordinates
