Theorems
This theorem is often useful when computing a line integral of a conservative vector field
What is the Fundamental Theorem for Line Integrals
This line integral differential is equivalent to the magnitude of the derivative of the parameterization of the given curve times dt
What is ds
This theorem is often useful when computing a surface integral of a vector field across a closed surface.
What is the Divergence Theorem?
Type of vector field when there exists some potential function f such that for a vector field F, F = del f
(imagine F is bolded)
What is a conservative vector field F<x,y>
(a1)(b1) +(a2)(b2) + (a3)(b3)
and
|a||b|cos(theta)
What is the dot product of the vectors a and b?
These descriptors are required of a curve in 2D-space in order to use Green's Theorem when computing a line integral of a vector field
What is positively-oriented, simple, and closed
Also acceptable: What is positively-oriented and consisting of finitely many simple, closed curves
(answer order matters)
What are mass and work
This differential represents the area of a very, very small patch of the surface S, and thus is computed using the magnitude of a cross product times dA.
What is dS?
This kind of physical field must be solenoidal; that is, must have a divergence of 0.
What is a magnetic field
This produces a vector that is perpendicular to both vectors a and b and the area of the parallelogram produced by the vectors a and b.
What is a x b?
This theorem related to partial derivatives may be useful when proving the cross partial test for vector fields in 2-space
What is Clairaut's Theorem
If the vector function r(t) = <x(t), y(t), z(t)>, then the differential vector dr is equal to this expression
What is <dx, dy, dz> or <x'(t), y'(t), z'(t)>dt
A surface S has this property if we can make a continuous choice of a unit normal vector for each point on S.
What is orientable?
This is the curl of a conservative vector field defined on a simply connected domain.
What is the zero vector
This takes of a scalar function to produce a vector field
for a scalar function f, it is equal to (if you imagine the ds as partial symbols):
<(d/dx)f, (d/dy)f, (d/dz)f>
What is the Gradient of f?
This theorem can help with finding the flux of a vector field across a surface S of a portion of a solid.
What is the Divergence Theorem?
Theorem for line integrals that allows for such an integral over a boundary curve to be computed as a double integral over the enclosed region.
What is Green's Theorem?
This theorem indicates that the surface integrals of the curl of a vector field across two piecewise-smooth, oriented surfaces sharing a simple, closed boundary curve are the same.
What is the Corollary to Stokes' Theorem
This vector function parametrizes the line segment from (1,1,1) to (-1,0,2) with a parameter domain of [0,1].
What is r(t) = <1-2t,1-t,1+t>?
This is calculated using the components of the gradient operator and the vector field
Represents the rotation of a vector field
What is Curl?
This condition on the vector field’s domain is not required for Stokes’ Theorem, even though a similar restriction is necessary for Green’s Theorem.
What is the domain being simply connected?
For a vector field F the line integral ∫c F⋅dr is path-independent if and only if this condition holds on a simply connected domain.
What is if the vector field F is conservative?
The units of mass flow rate, where density is measured in kilograms per cubic meter and velocity is measured in meters per second.
What is kilograms per second, per square meter?
The vector function r(u,v) = <cos(u)sin(v), sin(u)sin(v), cos(v)> describes this kind of surface.
What is a sphere of radius 1?
This is calculated by finding the dot product of the del operator and the vector field
measures "outflowing-ness" of a vector field
What is divergence?