Scalar and Vector Fields Flashcards
What is the difference between a Scalar and a Vector field?
- A scalar field is a function ϕ(x, y, z) that assigns a scalar value to each point (x, y, z) in
its domain D. - A vector field F(x, y, z) which defines a vector at each point (x, y, z) in its domain D.
eg.
F(x, y, z) = F1(x, y, z) e1 + F2(x, y, z) e2 + F3(x, y, z) e3 .
What are field lines?
Field lines are visualised as a collection of arrows with magnitude and direction with a specified point in space.
What must be satisfied for 2 vectors to be parallel?
Two vectors are parallel if and only if one is a multiple of the other.
What does a gradient act upon and produce?
Gradient (grad) acts upon a scalar to produce a vector.
What is the chain rule for gradients?
grad ϕ(u(x, y, z)) = ϕ′(u) ∇u.
What do the terms normal and orthogonal mean?
Perpendicular.
What does divergence act on and what does it produce?
Divergence acts upon a vector field and produces a scalar field.
What does curl act upon and what does it produce?
Curl acts upon a vector field and produces a vector field.
Define the laplacian.
The Laplacian, del squared or div(grad ϕ) of vector field ϕ is simply the divergence of the gradient of ϕ.
Where F is a vector field and f is a scalar field, answer the following.
div(curl F) = ?
curl(grad f) = ?
div(curl F) = 0.
curl(grad f) = 0.
if vector field F satisfies curl(F) = 0, what is F said to be? Similarly, if F satisfies div(F) = 0, what is this said to be? What can the sum of two vectors satisfying these two characteristics be defined as?
F, in curl(F) = 0, is said to be irrotational.
F, in div(F) = 0, is said to be solenoidal (or divergence free).
The sum of vectors satisfying these can produce any vector field F. (This is Helmholtz’s theorem, not expected to remember).
