Midterm 2 Study Flashcards
(20 cards)
How can you prove that a vector field, F, is conservative?
Curl F = 0
What is the formula for Curl
gradient cross F
What is formula for Divergence for function F
Gradient dot F
What does divergence tell us?
How much a vector field diverges, spreads out from a point
What does curl tell us?
How much a vector field gyrates, and forms swirls (eddies)
For a vector field, G, to exist such that curl G exits on the same plane, what needs to be satisfied?
Divergence ( Curl G) = 0
If the above function is not true, then vector field G does not exist.
What is a parametric surface?
Like a parametric equation except each component is a function of 2 variables
What do you need for the equation of a plane?
A normal vector and a point on the plane
What is the equation for a plane
(Normal vector) dot (r-r0)=0
Where r= <x,y,z> (general point )
And r0 is known point on the plane
How do you find the normal vector of a parametric surface at a point?
Find the tangent lines in both the u and v directions (where u and v are the variables used in the parametric surface equation) and cross product them. The tangent line in the u direction is the partial derivative of the parametric surface equation with respect to u.
What is the formula for the area of a parametric surface?
Double integral over the domain S of the magnitude of the normal vector (Ru cross Rv)
What is Greens theorem?
For a positively oriented curve that is closed and simply connected, the line integral of Pdx +Qdy = double integral over the domain of the closed loop of (dQ/dx -dP/dy) dA
What’s an extension of Greens theorem when you want to take the line integral of a function over the tangential component of a line (Work) (F dot dr) AKA (Pdx+ Qdy)
It’s equal to the double integral over the domain D, area enclosed by curve, of curl F
What’s an extension of greens theorem when you want to take a line integral of a function over a curve with respect to the normal of the curve (F dot dn)
= double integral over domain D, area enclosed by curve, of the divergence F
To change variables when solving double integrals what does the inside of the integrals change to
F(x(u,v), y(u,v)) (jacobian) DA
To change variables when solving double integrals what does the inside of the integrals change to
F(x(u,v), y(u,v)) (jacobian) DA
What is a Jacobian
Cross product of dx/d(u,v) by dy/(d(u,v))
Is F is a conservative vector field or can be written as..
The gradient of a potential function f
What is the fundamental theorem of line integrals
If F is conservative, then the line integral of F(r(t)) is the f(r(b)) - f(r(a)) where f is F’s potential function and the bounds are t goes from a to b
What is the parametric formula of a line? When given two points? r0 and r1
r(t) = (1-t)r0 + tr1
