Unit 5: Vector Calculus Flashcards
How do you compute the curl of a vector field?
curlF = ∇× F
(think curl and cross)
How do you compute the divergence of a vector field?
divF = ∇⋅ F
(think div and dot)
What does it mean for a vector field to be conservative?
If there’s a scalar function f(x,y) such that ∇f = F, then F is conservative (F is a vector field F(x,y))
Note: this also works in 3 dimensions (so if F(x,y,z) = ∇f(x,y,z) then F is conservative)
What is a potential function?
Given vector field F, if F = ∇f for some function f, f is the potential function of vector field F
Note: this is the one where you have to calculate integrals and the + C is really like a plus g(y) cuz you’re doing partial x
What does it mean if a vector field is conservative? (Not the definition of conservative vector field - there’s a second math fact to go with it)
If vector field F is conservative, then f is a potential function of F
Properties of Divergence and Curl 1 of 4:
What does it mean if curlF = 0?
If curlF = 0 then F is conservative
Properties of Divergence and Curl 2 of 4:
div(curlF) = ?
div(curlF) = 0
Can you take curl(divF)?
NO because divF returns a scalar (think div and dot) and curlF needs vectors (think curl and cross)
Properties of Divergence and Curl 3 of 4:
What does it mean if F is conservative?
If F is conservative then curlF = 0
Properties of Divergence and Curl 4 of 4:
curl(∇f) = ?
curl(∇f) = 0
Because ∇f = F; if you’re given the potential function then F must be conservative –> if F is conservative, curlF = 0
What three things relating to curl and divergence are significant if they equal 0? (some of them always equal 0)
Significant, have to check if = 0:
1. If curlF = 0 then F is conservative
ALWAYS equal 0
2. div(curlF) = 0
3. curl(∇f) = 0
How do you write the equation for work in calculus?
Work: ∫F ⋅ dr = ∫(Pdx + Qdy +Rdz) = ∫{from t=a to t=b} F(r(t)) ⋅ r’(t) dt
(∫ has a subscript of C for the F•dr one)
