Big List of Topics from Math 241 for possible FITB questions

Vectors and Distance
 -multiplication of vectors by scalars, vector+vector addition
 -tip minus base
 -length of vectors
 -unit vectors (directions)

Dot and Cross products of 3D vectors
 -component formulas
 -geometric formulas
 -meanings when zero (perpendicular, parallel)
 -interpretations (projection, work; torque, area)
 -volume interpretation of the absolute value of the triple product

Parametric and Nonparametric equations
 -lines
 -planes
 -distances between two figures (points, lines, planes)

Parameterized Curves (vector-valued functions of one real parameter)
 -C^k, k-times continuously differentiable, k=0,1,2,...
 -no cusp condition
 -unit tangent vector
 -arc length (differential and total)

Domain and Range Sets for real-valued functions n>=2 independent variables
 -domain (natural; specified), codomain, range
 -open set
 -boundary points of a set
 -closed set
 -bounded set
 -level sets of functions and contour diagrams

Limits of functions of n>=2 independent variables
 -definition (epsilon and delta)
 -limit exists, DNE
 -limit theorems (sum, scalar multiple, product, quotient)
 -sublimits (restricting to curves in the domain)
 -continuous function (at a domain point; at all domain points)
 -sums, products, compositions of continuous functions are continuous

Differentiability
 -partial derivatives
 -definition of differentiability in n>=2 variables
 -example where partial derivatives exist but function not differentiable
   -f(x,y)=sqrt(|xy|) at (x,y)=(0,0)
   -f_x(0,0)=0, f_y(0,0)=0, f not differentiable at (0,0)
 -C^k, k-th order partial derivatives exist and are continuous on domain
 -C^1 on open domain implies differentiability
 -C^2 on open domain implies equality of mixed partials e.g. f_xy=f_yx

Gradient
 -definition
 -Chain Rule statement (differentiability hypothesis)
 -directional derivatives (maximal, minimal), length of the gradient
 -interpreting the direction of the gradient
 -relation of the gradient at a point to the level set through that point
 -zero directional derivative along tangents to the level set

Tangent Planes (nonparametric equations for)
 -to graphs
 -to level surfaces

Double Integrals
 -definition (limits of sums as the decomposition of the domain becomes finer)
 -iterated integrals as a means to calculate double integrals
 -Fubini's Theorem over a rectangle (hypothesis of continuity)
 -Fubini's theorem over dxdy and dydx regions
 -polar coordinates, x=r cos theta, y=r sin theta, formulas for r and theta
 -dA=r dr dtheta
 -f(x,y)=f(r cos theta,r sin theta)
 -polar rectangles, dr dtheta and dtheta dr regions
 -A double integral of f(x,y) dA over a dr dtheta region can be written as 
    an interated integral of f(r cos theta,r sin theta) r dr dtheta

 Line Integrals
 -definition of int_C f ds over a parameterized curve C
 -orientations of a curve, C^# is an oriented curve
 -unit tangent vector T of an oriented curve
 -vector fields F=<P,Q> G=<M,N> (2D); or F=<P,Q,R> (3D)
 -definition of a flow-type line integral: int_C F dot T ds
 -alternate notations for flow-type line integrals
    - int_C^# F dot dr
    -(2D) int_C^# P dx+Q dy
    -(3D) int_C^# P dx+Q dy+R dz
 -reversing the orientation negates the flow-type line integral 
 -interpretation of flow-type line integrals: e.g. work
 -closed curves, concatenated oriented curves
 -circulation=flow type line integral over an oriented closed curve
 -right pointing unit normal vector Nvect for a 2D oriented curve
 -definition of flux-type line integrals in 2D: int_C G dot Nvect ds
 -alternate notation, flux of G=<M,N> across C^#: int_C^# M dy-N dx

Gradient Vector Fields
 -Fundamental Theorem of Line Integrals: int_C^# gradient f dot dr=f(end)-f(beg)

Green's Theorem (GT) 
 -the boundary curve C of a 2D region R
 -the boundary orientation on the boundary curve (C^#): stand on boundary, 
  face the positive direction of C^#, exterior side of R on right.
 -outward flux=flux across C^#
 -GT on a rectangle R (C^1 hypothesis on F, boundary orientation on C^#)
   -flow form: intint_R Q_x-P_y dA=int_{C^#} P dx+Q dy
   -flux form: intint_R M_x+N_y dA=int_{C^#} M dy-N dx
 -GT on more general regions (dxdy, dydx, drdtheta, dthetadr)
 -alternate notations for GT