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CAL / Math / MATH 53 / What are the limits of multivariable functions?

What are the limits of multivariable functions?

What are the limits of multivariable functions?


School: University of California Berkeley
Department: Math
Course: Multivariable Calculus
Professor: Edward frenkel
Term: Fall 2016
Tags: multivariable calculus
Cost: 50
Name: Math 53 Midterm 2 Study Guide
Description: Study Guide for Stankova's Spring 2018 Math 53 Multivariable Calculus Class.
Uploaded: 03/30/2018
13 Pages 15 Views 4 Unlocks

Midterm 2 Study Guide

What is the limits of multivariable functions?

Tuesday, March 27, 2018 1:23 PM

Limits of Multivariable Functions




the function is continuous:ConstantsSine CosinePolynomials+ ‐ * of these functions

the domain of f.Partial Derivatives

It is denoted, for a function f(x,y), fClairaut's Theorem says that if fcontinuous, then they are equal.

Linear Approximation



Partial derivatives refers to what?

Approach a limit from different paths

f the result for the limit varies, then there exists no limit You can just use the value of a function as the limit if you know 



Division and square root (of these functions) are 

continuous on their domains

Composition of functions: f(g(x)), assuming range(g(x)) is in   

A partial derivative is a derivative of an expression or function  of multiple variables. We treat everything but our desired  partial variable as constants and take the derivative as normal.  x or fy.

xy (taking (fx)y) and fyx are 

Linear approximation refers to what?


- f(x,y) must have continuous partials fx and fy near (x0, y0). - The linear approximation is: We also discuss several other topics like What is the name of telemachus’ mother?

 Cheat Sheets Page 1

○ L = f(x0,y0) + fx(x0,y0)(x‐x0) + fy(x0,y0)(y‐y0)

The Chain Rule 

- The basic chain rule learned in Calculus 1:

- We extend this to the multivariable case with partials:

- We can represent this using a tree diagram:  Cheat Sheets Page 2

 Cheat Sheets Page 3

- Verification.

To verify a solution to a given differential  equation, we attempt to find the quantities in the  differential equation and see if those are indeed  equal, as defined by the equation.

Directional Derivatives: Gradient 

- Definition of the directional derivative:

 Cheat Sheets Page 4

- Gradients are tangent to level curves of a function. Tangent lines and planes 

- Don't forget about the age old question of What are the assumptions, in monopolistic competition?

Tangent line to a curve can be thought of as a tangent  line to an equivalent level curve of a surface:

 Cheat Sheets Page 5


Find the gradient, and the point you want to be  tangent at.

- Since gradient is tangent to the level curve:


The general equation for a tangent line/plane at point  P is:

Local min/max of a function of two variables. - Combine the first and second derivative tests. - Second derivative of a function:

- If:

○ D > 0, fxx > 0: minimum

○ D > 0, fxx < 0: maximum We also discuss several other topics like What do we mean by self-esteem or addiction?

○ D < 0: saddle point

○ D = 0: unsure. Try a different test.

 Cheat Sheets Page 6

Finding Global Extrema

Very similar to global extrema in single variable. 


Check for critical points and then check the  endpoints. However, with more dimensions, our  endpoints may be actual equations or even surfaces. -

A bounded region has a boundary, or it can be  contained within a shape of finite size. Don't forget about the age old question of What is the difference between essential and non-essential of amino acids?
If you want to learn more check out Who uses water?

- A close region is a region that contains its boundary. - Finding the global extrema

Find the critical points of the function, where the  gradient of f = <0,0> or <0,0,0> If you want to learn more check out What does the law of superposition?

Calculate equations for the boundaries of f, or use  Lagrange multipliers.

With equations, find their min/max with critical  points, and find their endpoints as well. ▪

With Lagrange multipliers, solve this system of  equations:

 Cheat Sheets Page 7

○ Then, compare all your points

Note. Lagrange multipliers doesn't always solve  for all the min/max on a region. You have to  checks spots where the gradients or the  derivatives are discontinuous.

Multiple Integrals 

 Cheat Sheets Page 8


Approximations are the same with these, but we  simply test with more points because there are extra  dimensions.

Iterated Integrals 

 Cheat Sheets Page 9

Polar coordinate integrals 

- You can do the same integral, substituting properly:  Cheat Sheets Page 10

But, you have to multiply your f(r,theta) by r after 


substituting. This is the Jacobian, a factor based on  how you actually changed the area when you  changed to polar coordinates.

Spherical coordinates integrals 

- Spherical coordinates:

 Cheat Sheets Page 11

Change of Variables, change of basis 


When changing from x, y to u, v to calculate some  integral, we need to find the Jacobian for that new  space. This is what we've done for polar and for  spherical coordinates.


Find x = f(u,v)

        y = g(u,v)

- Compute:

 Cheat Sheets Page 12


The Jacobian for more variables extends to the  determinant of a larger matrix (3 by 3 for three, 4 by 4  for four, etc.)

 Cheat Sheets Page 13

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