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THE CHINESE UNIVERSITY OF HONG KONG

DEPARTMENT OF MATHEMATICSMAT2010A (First Term, 20102011)

Advanced Calculus INotes 1 Multivariable Preview

1.1 Review

There is certain similarity in the study of 1-variable and multivariable differential calculus. For

this reason, it is beneficial for the readers the refresh their knowledge of 1-variable calculus.

1.1.1 Overview

Try to come up with a half-page outline of the major content of 1-variable differential calculus.

Besides the topics, think about what and how do we study them.

1.1.2 Concept and skills

Here are some exercises, which may be considered the pre-requisite of learning multi-variable

calculus.

1. Let R be the set of real numbers and A R. What is the meaning of an upper boundofA and sup A? Can you give the definition in set language?

2. Does the limit limt0

t+ 1/tt 1/t exist?

3. How do you define that a functionfis continuous at x = a?

4. Find the derivative offwherever it is well defined,

f(x) =

3x2, x 1 ,2x3 + 1, x >1 .

5. State various versions of the Mean Value Theorem, and the Taylor Expansion. Do you

know how they are used in proving the LHospitals Rule?

6. Sketch the graph of the function

f(x) = x2 sin x

x2 2.

7. Write down the procedures of finding the absolute maximum or minimum of a function

over a closed interval. Justify every step.

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1.2 Preview

The following is a comparison of the outlines one variable differential Calculus and multivariable

differential Calculus. It may not be rigorously presented, but it does show the general similarity

between the two.

1-variable multivariable

Object function y = f(x) z = f(x, y) or

z = f(x1, . . . , xn) =f(x) or

(z1, . . . , zm) =f(x1, . . . , xn)

Picture Graph in R2

y = f ( )x

Graph in R3 or

Graph Rm+n in generalLines, planes, and hyperplanes

standard surfaces in R3

Analysis limit limit (harder)

derivative partial or directional derivatives

differentiability??

higher derivative higher partial derivatives

changing order of differentiations

Geometry tangent line tangent line and plane

Existence of tangent plane??

curve sketching visual understanding

gradient and level curves

byf and f by gradient and hessian

eigenvalues of hessian matrix

Applications rate of change differential

linear approximation

max / min problems max / min problems

byf = 0 and f by gradient =0 and hessian

eigenvalues of hessian matrixnegative / positive definite

While there are similarities, there are also differences. The difference is not only easier versus

harder. At certain places, there is fundamental difference because of the number of variables.

For example, in single variable, a function is differentiable if and only if its derivative exists.

However, the existence of partial derivatives does not guarantee differentiability in multiple

variables. Generally speaking, the difference of the theories lies on n = 1 versus n 2, where nis the number of independent variables.

Next. Assume you understand what R is, for the purpose of doing calculus, think about the

most crucial properties ofRn

.

Thomas Au MAT2010A Notes 1: Multivariable Preview 2