3) Line, Surface and Volume Integral Flashcards
What is a line integral in the context of vector fields
A line integral measures the integral of a vector field along a curve
How are line integrals computed in both two and three dimensions
What is a Conservative Vector Field
- f is known as a conservative vector field if it can be written as - f = ∇Φ
for some scalar function Φ, often called a potential - f is a conservative vector field in a domain D then -
∇ × f = 0
What is the Jacobian of a transformation between coordinate systems (2D)
What is Fubini’s Theorem
What is Stoke’s Theorem
What is Green’s Theorem
How are Stokes’ Theorem and Green’s Theorem equivalent
What is a line integral using the normal vector
What is Gauss’ (divergence) theorem
How is a triple integral defined in using Cartesian coordinates
How are triple integrals computed in curvilinear coordinates
How is the volume element represented in curvilinear coordinates using the Jacobian determinan
What are surface integrals of scalar fields and how are they evaluated on surfaces in 3D
Surface integrals of scalar fields involve integrating a scalar function u over a two-dimensional surface σ embedded in three-dimensional space.
What are surface integrals of vector fields
Surface integrals of vector fields calculate the flux of a vector field u through a surface σ
How are surface integrals of scalar fields calculated by parametrisation
How are surface integrals of scalar fields calculated by projection
Describe the surface orientation of a closed surface
For a closed surface σ:
* Positive Orientation: The normal vector points outwards from the surface,
* Negative Orientation: The normal vector points inwards towards the center of the solid body
How are surface integrals of vector fields calculated by parametrisation
How are surface integrals of vector fields calculated by projection
What is Stoke’s Theorem in 3D
