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Question about Calculus III

  1. Apr 10, 2008 #1
    If Calculus I is differential Calculus and Calculus II is Integral Calculus then what does calculus III entail.
  2. jcsd
  3. Apr 10, 2008 #2


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  4. Apr 10, 2008 #3


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    Yes, "Multivariable", as Vid said. Check the course descriptions in the college catalog. Calc 3 covers multiple integrations, partial derivatives, some more things about series and sequences, vectors.
  5. Apr 10, 2008 #4


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    IMO Taylor Series is the most important thing covered in Cal3
  6. Apr 10, 2008 #5
    Multivariable/Vector Calculus essentially

    Taylor series is covered in calc 2
  7. Apr 10, 2008 #6


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    My school didn't cover it until Cal3... maybe it's different in Canada.
  8. Apr 10, 2008 #7
    Ah, yeah maybe^^

    In the US it's typically covered at the end on calc II

    A typical course looks like:
    The 3-D Coordinate System
    Equations of Lines
    Equations of Planes
    Functions of Several Variables
    Vector Functions
    Calculus with Vector Functions
    Tangent, Normal and Binormal Vectors
    Arc Length with Vector Functions
    Cylindrical Coordinates
    Spherical Coordinates
    Partial Derivatives
    Interpretations of Partial Derivatives
    Higher Order Partial Derivatives
    Chain Rule
    Directional Derivatives
    Tangent Planes and Linear Approximations
    Gradient Vector, Tangent Planes and Normal Lines
    Relative Minimums and Maximums
    Absolute Minimums and Maximums
    Lagrange Multipliers
    Double Integrals
    Iterated Integrals
    Double Integrals over General Regions
    Double Integrals in Polar Coordinates
    Triple Integrals
    Triple Integrals in Cylindrical Coordinates
    Triple Integrals in Spherical Coordinates
    Change of Variables
    Vector Fields
    Line Integrals
    Line Integrals of Vector Fields
    Fundamental Theorem for Line Integrals
    Conservative Vector Fields
    Green’s Theorem
    Curl and Divergence
    Parametric Surfaces
    Surface Integrals
    Surface Integrals of Vector Fields
    Stokes’ Theorem
    Divergence Theorem
    Last edited: Apr 10, 2008
  9. Apr 11, 2008 #8


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  10. Apr 11, 2008 #9


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    there are two main ideas:
    1. in differentiation of two variable functions f, the main point is the gradient vector gradf, which lies in the domain of f, and points in the direction of greatest increase of f, hence at any point p, is perpendicular to the "level set" passing through p, i.e. the set where f has the same value as at p. the components of gradf are "partial" derivatives (fx,fy) computed by considering f as a function first only of x, then only of y.

    2. in integrating functions of two variables, the main idea is that of volumes by slicing, or cross sections, which means you can reduce a double integral to the successive computation of two single integrals.

    when looked at closely, this also can be phrased as implying that the integral measuring the flow of a "vector field" with components (M,N) around the boundary curve of a closed region R, equals the double integral of Nx-My taken over R, (where Nx and My are partial derivatives of N,M wrt the variables x,y).

    that is called greens theorem.

    thus the subject consists in defining analogs in several variables of the old friend ideas: namely derivatives and integrals; i.e. tools for measuring how a function changes, and for averaging its values; and then giving techniques for reducing the calculations of these concepts to the old calculations in one variable.

    a great technical convenience is acquired by studying how these calculations change when we change variables. for derivatives these are called chain rules, and for integrals are called change of variable formulas for integrals. the chain rule for derivatives basically says the deriv of a composite is computed by taking dot products, or matrix products of the successive derivatives. for integrals, volume changes by multiplying by the determinant of the linear approximation of the transformation.

    e.g. polar coords or cylindrical coords, or spherical coords are useful when studying objects of those shapes. in my class i also taught elliptical coords but these are not usually mentioned. still computing the volume of an ellipsoid is easiest in elliptical coords, just as computing volume of a sphere is easiest in spherical coords.

    i have covered most of the topics listed in post 7 above. (since those topics listed after green are just higher dimensional, or rotated, or curved, versions of green.)
    Last edited: Apr 11, 2008
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