# Ch14Notes - MATH 214: PARTIAL DERIVATIVES 1. Functions of...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 214: PARTIAL DERIVATIVES 1. Functions of Several Variables Definition. Suppose D is a set of n-tuples ( x 1 ,x 2 ,...,x n ) (usually n = 2 or 3 for us), where x i ∈ R . A real-valued function on D is a rule f that assigns a number w = f ( x 1 ,...,x n ) to every ( x 1 ,...,x n ) ∈ D . The set D is called the domain of f and the set { f ( x 1 ,...,x n ) | ( x 1 ,...,x n ) ∈ D } ⊂ R is called the range of f . The variable w is called the dependent variable of f and the variables x i are the independent variables of f . Example 1. The domain of f ( x,y,z ) = z √ xy is { ( x,y,z ) | xy 6 = 0 } = { ( x,y,z ) | x 6 = 0 } ∩ { ( x,y,z ) | y 6 = 0 } and the range is R . Example 2. The domain of g ( x,y ) = arctan x y is { ( x,y ) | - π 2 < x < π 2 ,y 6 = 0 } Definition. (1) Given a function f ( x,y ) , the set { ( x,y,z ) | z = f ( x,y ) } ⊂ R 3 is called the graph of f . For the functions f that we will consider, this will usually be a surface. (2) The set { ( x,y ) | f ( x,y ) = c } ⊂ R 2 is called the level curve of f at c . For the functions that we will consider, this will usually be a curve in the xy-plane. Example 3. The level curves of f ( x,y ) = x 2 + y 2 at c are circles of radius √ c if c > and are empty (i.e. they have no points) if c < . Example 4. The level curves of the function f ( x,y ) = xy at c are hyperbolas if c 6 = 0 and the level curve of f ( x,y ) = xy at c = 0 is the set { ( x,y ) | xy = 0 } = { ( x,y ) | x = 0 } ∪ { ( x,y ) | y = 0 } , i.e. the union of the x-axis and the y-axis. 2. Limits Definition. We say that f ( x,y ) approaches the limit L ∈ R as ( x,y ) approaches ( x ,y ) if for every > , there exists a δ > such that for every ( x,y ) with < p ( x- x ) 2 + ( y- y ) 2 < δ we have | f ( x,y )- L | < . In this case we write: lim ( x,y ) → ( x ,y ) f ( x,y ) = L Example 5. lim ( x,y ) → ( x ,y ) x = x , lim ( x,y ) → ( x ,y ) y = y , lim ( x,y ) → ( x ,y ) k = k for any k ∈ R . Rather than calculate limits directly, we’ll usually refer to the following theorem: Theorem 2.1. If lim ( x,y ) → ( x ,y ) f ( x,y ) = L and lim ( x,y ) → ( x ,y ) g ( x,y ) = M , then: (1) lim ( x,y ) → ( x ,y ) c 1 f ( x,y ) + c 2 g ( x,y ) = c 1 L + c 2 M (i.e. limits are linear ) 1 2 MATH 214: PARTIAL DERIVATIVES (2) lim ( x,y ) → ( x ,y ) f ( x,y ) g ( x,y ) = LM (this is sometimes called the product rule ) (3) lim ( x,y ) → ( x ,y ) f ( x,y ) g ( x,y ) = L M if M 6 = 0 (the quotient rule ) (4) If the greatest common denominator of r and s is 1 and s 6 = 0 , then lim ( x,y ) → ( x ,y ) f ( x,y ) r s = L r s (called the power rule ). For the power rule, we assume that L > if s is even (i.e. we don’t take even roots of negative numbers)....
View Full Document

## This note was uploaded on 07/31/2011 for the course MATH 214 taught by Professor Alexondrus during the Spring '11 term at University of Alberta.

### Page1 / 9

Ch14Notes - MATH 214: PARTIAL DERIVATIVES 1. Functions of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online