Final Review

Analytic Geometry

  • Points: (x,y), (x,y,z)
  • Distance between two points
  • mid-point
  • vectors in 2D and 3D: <x,y>, <x,y,z>
  • vector formed from two points, direction
  • vector operations and geometric and physical meanings:
  • a + b,  a - b,  a dot b, a cross b. Parallelogram and triangle laws, work, area, collinear of three points
  • Triple product, volume of ?
  • lines in 2D and 3D, different forms: parametric, symmetric, component forms. Line between two points
    • A point + a direction
  • curves in 2D and 3D, different forms, parametric forms of curves
  • Special curves: e.g. circles, ellipses, helix,

    • x=a cos t,  y=b sint
  • parametric form of plane curves:
  • y = f(x)
  • x=f(y)
  • f(x,y) = 0
  • tangent direction
  • arc-length, ds, dr, dx, dy,dz
  • The equation of the tangent line at a point.
  • curvature*
  • planes in 3D
  • a point + normal direction
  • three points
  • two lines
  • parametric form
  • surfaces in 3D and sketch them:
  • Single equation f(x,y,z)=0, parametric form?
  • graph of g(x,y): z=f(x,y), x=g(y,z), y=h(x,z), parametric form.
  • quadric surfaces and parametric forms:
  • sphere: center+radius, how to tell, complete square
  • ellipsoid, paraboloids, hyperboloid, cones, cylinders
  • ds and dS,  the surface area and the volume below the surface
  • The equation of the tangent plane at a point
  • Express surfaces in cylindrical and spherical coordinates

    • Calculus: functions, differentiation and integration

    Functions in 2D: f(x,y)

  • Sketch domain, range, the graph, traces
  • sketch of the level curves, what is the difference between traces and level curves?
  • limits
  • regular situations
  • weird situations and y=k x approach
  • continuity
  • partial derivatives
  • The equation of the tangent plane of the graph
  • high order derivative
  • chain rule, z=f(x,y),  x=x(u,v), y=y(u,v), find zu and zv.
  • differentials and approximation, Taylor expansion at a point
  • implicit function, e.g. given F(x,y,z)=0, find zx and zy.
  • FIND ALL CRITICAL POINTS and CLASSIFY THEM
  • Second order derivative test: D=fxx f yy - (fxy)2
  • D>0, local minumum/maximum
  • fxx > 0, local minimum
  • fxx <0 , local maximum
  • D <0, a saddle point
  • Find the global extremes

    •  
  • DOUBLE INTEGRALS
  • Geometric meaning:
  • Volume between the surface z= f(x,y) and xy plane
  • Area of the domain if f=1.
  • Iterated integrals
  • Fubuni's theorem if the domain is a rectangle
  • Between two curves and the line test. Which variable should we integrate first?
  • BREAK-UP
  • POLAR coordinates
    • x = r cos t
    y = r sin t                    d A = dx dy = r dr dt
  • Application of double integrals: Volume, area, mass, center, ....
  • TRIPLE  INTEGRALS
  • Geometric meaning:
  • Volume of the domain  if f=1.
  • Iterated integrals
  • Fubuni's theorem if the domain is a cubic
  • Between two surfaces and the line test. Which variable should we integrate first?
  • BREAK-UP
  • Special coordinates
    • x =
    y =                    d V = dx dy dz= ?
    z =
    What are the meanings of each new variables? Try to express the sphere in different parts of the space
  • Cylindrical coordinates (get rid of one variable + polar coordinates)
  • Application of triple integrals: Volume, area, mass, center, ....
  • VECTOR FUNCTION and CALCULUS
  • Magic gradient operator
  • gradient f(x,y),  gradient f(x,y,z)
  • divergence
  • curl
  • Vector function is 2D: F = <P, Q>, 3D: F = <P, Q, R>,
  • line integral with f(x,y) ds, and  F(x,y) d r
  • Parametric forms of plane and space curves
  • When line integral is independent of path?
  • When is F  conservative? Px = Qy in 2D;     curl F = 0 in 3D.
  • How do we take advantage if F  is conservative?
  • simple path
  • potential function
  • How do we find the potential function 1n 2D and 3D?
  • How do we integrate   grad f dot d r
  • Green's theorem (2D) of the line integrals
  • Closed plane curve and in positive direction
  • Use the Green's theorem to find the area of the domain, change to the line integral
  • Surface and flux integral
  • The parametric form of different surfaces
  • What are dS and dS? How do we find them?
  • Determine the domain of the parameter u and v
  • Change to double integral
  • Use Cartesian or polar coordinates?
  • Divergence Theorem
  • Close surface and outward direction
  • Change to triple integral

    • EXCLUSIONS:

    13.7, 12.9, 11.8, 10.4