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## Main Question or Discussion Point

If Calculus I is differential Calculus and Calculus II is Integral Calculus then what does calculus III entail.

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If Calculus I is differential Calculus and Calculus II is Integral Calculus then what does calculus III entail.

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Multivariable

symbolipoint

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nicksauce

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IMO Taylor Series is the most important thing covered in Cal3

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Multivariable/Vector Calculus essentially

Taylor series is covered in calc 2

Taylor series is covered in calc 2

nicksauce

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My school didn't cover it until Cal3... maybe it's different in Canada.

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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

Curvature

Cylindrical Coordinates

Spherical Coordinates

Limits

Partial Derivatives

Interpretations of Partial Derivatives

Higher Order Partial Derivatives

Differentials

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

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

Curvature

Cylindrical Coordinates

Spherical Coordinates

Limits

Partial Derivatives

Interpretations of Partial Derivatives

Higher Order Partial Derivatives

Differentials

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

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mathwonk

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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.)

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.)

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