Calculus 3 Differentiation and Integration of Functions of Many - PowerPoint PPT Presentation

Calculus 3 Differentiation and Integration of Functions of Many Variables for Physical Sciences and Engineering Hu` ynh Qang V˜ u Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City 05/2014

Vector functions Definition A vector function is a map f : D → R 3 where D ⊂ R . In the standard basis of R 3 we can write r ( t ) = � x ( t ) , y ( t ) , z ( t ) � .

Limits Te notion of limit can be generalized to any space with distance. Theorem t → a r ( t ) = � lim lim t → a x ( t ) , lim t → a y ( t ) , lim t → a z ( t ) � . Definition A function r is said to be continuous at t = a if lim t → a r ( t ) = r ( a ) . Fact A vector function is continuous if and only if each of its components is continuous.

Derivative Definition r ( t + h ) − r ( t ) r ′ ( t ) = lim , h h → 0 provided the limit exists. Te derivative represents the rate of change of r ( t ) with respect to t . Te derivative is the limit of secant vectors, therefore it is called a tangent vector . Physically, r ′ ( t ) is the velocity at time t .

Theorem If r ( t ) = ( x ( t ) , y ( t ) , z ( t )) then r ′ ( t ) = ( x ′ ( t ) , y ′ ( t ) , z ′ ( t )) .

Space curves Definition A space curve C is the set of all points r ( t ) , t ∈ D ⊂ R . r is called the position function or a path. C is said to be parametrized by r . Tus a curve is the trace of a path. Example Te curve given by the following parametrization x ( t ) = cos t , y ( t ) = sin t , z ( t ) = t is called a helix. Fact A curve can be parametrized by different functions.

Smooth curves Definition A curve is called smooth , or regular if there is a parametrization by r ( t ) , t ∈ D such that r ′ ( t ) is continuous and r ′ ( t ) � = 0 on the interior of D . With a regular parametrization, the speed is never zero, and the tangent direction is defined as the direction of the velocity vector.

Arc length Let r : [ a , b ] → R n be a smooth path. Consider a partition a = t 0 < t 1 < · · · < t n = b of [ a , b ] . On each interval [ t i − 1 , t i ] , 1 ≤ i ≤ n , linearly approximate the path: r ( t ) ≈ r ′ ( t i − 1 )( t − t i − 1 ) . Te “length” of the path r ( t ) from t i − 1 to t i is linearly approximated by the tangent vector r ′ ( t i − 1 )∆ t i . Tus the “length” of the path on [ a , b ] is approximated by n � | r ′ ( t i − 1 ) | ∆ t i . i = 1 Tis is exactly a Riemann sum of the function | r ′ ( t ) | on [ a , b ] .

Definition Te length of a smooth path r : [ a , b ] → R n is defined as ˆ b | r ′ ( t ) | dt . a The length of a path is the integral of its speed with respect to time . It reminds the old formula: distance=speed x time. Example Suppose that an object is traveling on a line from time a to time b with constant speed v > 0. Ten the distance it has traveled is ´ b a v dt = v ( b − a ) , as we expect.

Arc length function Theorem Let r : [ a , b ] → R n be a regular path. Te function ˆ t | r ′ ( u ) | du . s ( t ) = a is called the arc-length function of r. s ( t ) has an inverse function t ( s ) , 0 ≤ s ≤ l , where l = s ( b ) is the length of r . Te path β ( s ) = r ( t ( s )) has the same trace as r . Te speed of β is always 1. β is said to be the re-parametrization by arc-length of r . Note that ds dt ( t ) = | v ( t ) | . Symbolically: ds = | v ( t ) | dt .

Curvature Te curvature of the curve at a point p is a measure of how fast the direction of the curve turns around p . It is therefore a measure of the rate of change of the direction of the curve as one moves along the curve at constant speed. Te direction of the curve is given by the unit tangent vector T ( t ) = r ′ ( t ) | r ′ ( t ) | . Te constant speed is given by parametrization by arc-length. Definition Let r : [ a , b ] → R n be a regular path and let C be its trace. Te curvature of the curve C at a point p = r ( s ) is defined as the number � � dT � � k ( p ) = ds ( s ) � . � � � � � Tus if the curve r ( t ) has constant speed then the curvature is exactly | r ′′ ( t ) | .

Usually the curve is not given in arc length form. For curve � � � T ′ ( t ) � � � � � � � � � � dt · 1 1 dT dT dt · dt dT dT � � � � � � � � � k ( p ) = � = � = � = dt · � = | r ′ ( t ) | . � � � � � � � � ds ds ds | r ′ ( t ) | � � � � � � � � � � � dt � Because T ( t ) = r ′ ( t ) / | r ′ ( t ) | , with some calculations we get a convenient formula: k ( t ) = | r ′ ( t ) × r ′′ ( t ) | | r ′ ( t ) | 3

In particular, for plane curve r ( t ) = ( x ( t ) , y ( t )) k = | x ′′ y ′ − x ′ y ′′ | ( x ′ 2 + y ′ 2 ) 3 / 2 . For a graph y = f ( x ) : | f ′′ ( x ) | k = ( 1 + f ′ ( x ) 2 ) 3 / 2 .

Limit Definition Suppose that f ( x , y ) is defined on D ⊂ R 2 . Let ( a , b ) ∈ D . We say that the function f has limit L as ( x , y ) approaches ( a , b ) , and write lim ( x , y ) → ( a , b ) f ( x , y ) = L if f ( x , y ) is arbitrarily close to L when ( x , y ) is sufficiently close to ( a , b ) . Precisely, for any ǫ > 0 there is a number δ > 0 such that if | ( x , y ) − ( a , b ) | < δ then | f ( x , y ) − L | < ǫ .

Continuity Definition We say that f is continuous at ( a , b ) if ( x , y ) → ( a , b ) f ( x , y ) = f ( a , b ) . lim Elementary functions, such as polynomials, are continuous.

Example In this case the point ( 0 , 0 ) is not in the domain of the function. x 2 − y 2 lim x 2 + y 2 ( x , y ) → ( 0 , 0 ) Te definition of limit needs to be extended for this case. Example Te main difficulty when evaluate limits is that there are infinitely many directions to approach a point on the plane , unlike the case on the line (there are only two directions). If two different directions give different limits then limit does not exist. In the above example, the direction x = 0 and the direction y = 0 give different limits − 1 and 1, therefore limit does not exist.

Partial Derivatives Suppose that ( a , b ) is a point in the interior of the domain of a real function f ( x , y ) . Fix y = b then f ( x , y ) is a function of x only. Can take the derivative with respect to x at x = a . Call it the partial derivative of f with respect to x at ( a , b ) : � � ∂ ∂ x f ( a , b ) . So ∂ f ∂ x ( a , b ) is the rate of change of f at ( a , b ) with respect to x . Other notations: � � ∂ ( a , b ) = ∂ f ∂ x ( a , b ) = f x ( a , b ) = D 1 f ( a , b ) . ∂ x f

Geometric meaning of partial derivatives When y = b is fixed, we get a path ( x , b , f ( x , b )) with parameter x on the graph ( x , y , f ( x , y )) of z = f ( x , y ) . Te trace of that path is the curve which is the intersection between the plane y = b and the graph z = f ( x , y ) . Te velocity of this path is the derivative with respect to x at x = a , which is ( 1 , 0 , f x ( a , b )) .

Higher partial derivatives ∂ f ∂ x is a function of ( x , y ) therefore we can talk about its partial derivatives. ( a , b ) = ∂ 2 f ∂ � ∂ f � ∂ x 2 ( a , b ) = f xx ( a , b ) = D 1 , 1 ( a , b ) . ∂ x ∂ x ( a , b ) = ∂ 2 f ∂ � ∂ f � ∂ y ∂ x ( a , b ) = f yx ( a , b ) = D 2 , 1 ( a , b ) . ∂ y ∂ x Theorem ∂ 2 f ∂ 2 f If ∂ y ∂ x and ∂ x ∂ y are continuous on an open set then they are equal there: ∂ y ∂ x = ∂ 2 f ∂ 2 f ∂ x ∂ y .

Proof. Idea: f y ( a + h , b ) − f y ( a , b ) f xy ( a , b ) = lim h h → 0 f ( a + h , b + k ) − f ( a + h , b ) − [ f ( a , b + k ) − f ( a , b )] = lim h → 0 lim hk k → 0 f ( a + h , b + k ) − f ( a , b + k ) − [ f ( a + h , b ) − f ( a , b )] = lim k → 0 lim hk h → 0 = f yx ( a , b ) .

Proof. Rigorously, let ∆( h , k ) = f ( a + h , b + k ) − f ( a + h , b ) − f ( a , b + k ) + f ( a , b ) . Let g ( x ) = f ( x , b + k ) − f ( x , b ) , then by the Mean Value Teorem there is α between a and a + h and β between b and b + k such that ∆( h , k ) = g ( a + h ) − g ( a ) = g ′ ( α ) h = [ f x ( α, b + k ) − f x ( α, b )] h = f yx ( α, β ) hk Similarly we get ∆( h , k ) = f xy ( α ′ , β ′ ) hk . From this: ∆( h , h ) lim = f xy ( a , b ) = f yx ( a , b ) . h 2 h → 0

Linear Approximation Suppose that f ( x , y ) is differentiable in a neighborhood of ( a , b ) . Let r ( x , y ) = ( x , y , f ( x , y )) . Te trace of r ( x , y ) is the graph of f . Fix y = b then r ( x , y ) becomes a path on the graph of f . Its velocity vector is r x ( a , b ) = ( 1 , 0 , f x ( a , b )) . Tis vector is “tangent” to the graph of f at the point ( a , b , f ( a , b )) . Similarly, fixing x = a we have another tangent vector r y ( a , b ) = ( 0 , 1 , f y ( a , b )) . Te two tangent vectors span a plane, called the tangent plane of the graph of f at the point ( a , b ) .

Tangent plane Tis tangent plane has a normal vector r x ( a , b ) × r y ( a , b ) = ( − f x ( a , b ) , − f y ( a , b ) , 1 ) . Terefore an equation for the tangent plane is: − ( x − a ) f x ( a , b ) − ( y − b ) f y ( a , b ) + ( z − f ( a , b )) = 0 . Te main idea of linear approximation is to use the tangent plane to approximate the graph . Tus for ( x , y ) “near” to ( a , b ) : f ( x , y ) ≈ f ( a , b ) + f x ( a , b )( x − a ) + f y ( a , b )( y − b ) . Or ∆ f ( x , y ) ≈ f x ( a , b )∆ x + f y ( a , b )∆ y .