multivariable calculus 14.4 – 14. 7 name - … multivariable calculus, continuous functions...

Multivariable Calculus 14.4 – 14. 7 Name:

Chain Rule, Gradient, Tangent Planes and Saddle Points

The Chain Rule The proof of this theorem is involved and very analytic, so we will give a quick outline of the proof. First, we must recall the definition of differentiability of a multivariable function.

If f is differentiable, then for every 1 2 and 0 there exis , 0 such tts hatx y

0 0 0 0( , ) ( , ) 1 2| |x y x y

f ff x y x y

x y

Dividing everything by t , we get

0 0 0 0( , ) ( , ) 1 2| |x y x y

f f x f y x y

t x t y t t t

As 0 we hope the epsilons dissappeart , and we get the chain rule theorem.

The following are various forms of chain rule:



Directional Derivatives and Gradient Vectors We know that the partial of a function, F, with respect to x is the derivative of F in the direction of x. If we are finding the derivative in the direction of a vector u in the domain of F, we can use parameterized

directed line segment in the direction of 1 2 0 0, ) from the point( ( , )u xu yu

0 1 2( ) ( , )osus y ur x s

will start at 00( , )x y and end at 1 2 )( ,u u u .



By the chain rule,

1 2

)·( ,

x y

x y

dF F dx F dy

ds x ds y ds

u F u

F

F

uF

We call the vector of partials the gradient.

Now that we have this language, we can compute the directional derivative more easily.

The following box contains the most important information you will ever learn about the gradient.

To summarize, the box above states that the gradient always points in the direction of greatest increase, and in the opposite direction of greatest decrease. It also says that if a direction is orthogonal to the gradient is pointing in the direction in which there is no change.

,( )F F

x y



Implicit Function Theorem Suppose ( , ) is differentiable and ( , ) 0 defines as a function of .F x y F x y y x

Since this is a level curve of F, then there is zero change in any direction, u, tangent to the level curve.

If this curve is the the graph of a function y of x, then dy

dx is tangent to the curve, and, hence, orthogonal

to the gradient, ( , )x yF F . This means that the slope of the tangent line to y as a function of x is

x

y

Fdy

dx F

The conditions for this to hold are that the partials are continuous and the 0F

y

in the region.

One can generalize the consequence above to a region containing the pre-image of any point and functions in any number of variables. This is because the pre-image of any point would be a level curve (or surface), and by definition, there would be no change in the F as you travel along any tangent to a level curve (or surface).

For three variables,

On the next page are some algebraic properties of gradient.



As you can see, the gradient acts like a derivative and, therefore, inherits the properties of the derivative.



Tangent Planes and Differentials

For a differentiable function in two variables, we can find the tangent plane at a point by analyzing the geometric interpretation of directional derivatives. The partial derivative with respect to x, is the change in the function value, ( , )z F x y as x changes. Therefore, the vector in the direction of this change is

given by )(1,0,F

x

and the vector in the direction of the change in F as y changes is given by the vector,

)(0,1,F

y

. These vectors are in the tangent plane, so to find the equation of the tangent plane, we need to

find a vector normal to these two vectors. To do that, we find the cross-product

1 0 (

0 1

, ,1)F F F

x x yF

i j

y

k

Therefore the equation for a tangent plane at a point is given by



This can easily be generalized



Estimating Change in a Specific Direction When we estimated change in single variable calculus, we used

0

dyy y x

dx

In two variables, you will need to estimate with and and the partials of withf y x f respect to each variable, x and y.

To estimate the change in f in the direction of u, we use the dot product of the gradient and u.



We can also use linearization to approximate the value of a function, just like we did with single variable differentiable functions using tangent lines. Remember how we used the tangent line of a function at a

point, 0x , to estimate the value of a function at a point near 0x .



Okay, so that’s enough linear approximation. It’s ugly and analytical, and we never really liked it anyway. But here are a couple of examples, in case you were interested.



Extreme Values and Saddle Points

The extreme value theorem in single variable calculus said that every continuous function achieved it’s maximum and minimum on a closed interval. In multivariable calculus, continuous functions achieve their maxima’s and minima’s on closed, bounded domains. Just like in single variable calculus, a continuous function on a closed and bounded domain will reach its extrema on the boundary or where the partial derivatives are zero or fail to exist.



Unlike with single variable calculus, where the first derivative test required you to take the first derivative of a function and find its zeros or where it didn’t exist, in multivariable calculus, a local extremum occurs where the tangent plane is horizontal. How would you test for this? A horizontal tangent plane would mean that it’s parallel to the xy-plane, or normal to the z-axis. In other words, the partial derivatives must be zero.



Saddle Points Saddle points are the multivariable analogy to inflection points for single variable calculus. However, the second derivative test is not as easy as identifying where the second derivative changes sign. The first problem is that there are four second derivatives. So below is the formal definition of saddle point. After you read it, we will analyze it to see what it really means.

As you can see, regardless of how close we get to ( , , ( , ))a b f a b , for every open disk centered at (a,b),

there will always be a point whose value is less than ( , )f a b , there exists a point ( , )x y where

( , ) ( , )f x y f a b , and a point whose value is greater than ( , )f a b , there exists a point ( , )x y where

( , ) ( , )f x y f a b . Therefore, ( , , ( , ))a b f a b is a saddle point.I’m sure you can also see that finding

saddle points this way would be very inefficient. So we need to find a faster way to identify saddle points.

f (x, y) > f (a, b)

f (x, y) < f (a, b)



In order to understand this theorem, let us recall the original method of computing the partial of f with respect to x. We took a plane parallel to the xz-plane at a fixed y-coordinate, and took the derivative of f in

the x-direction. We can, therefore, consider, xxf to be the concavity of that plane with the surface. In this

way, we can revert to the second derivative test for single variable calculus. If the second derivative is negative, then the curve is concave down, and, hence the critical point is a maximum. Similarly, if the second derivative is positive the critical point is a minimum. However, we still haven’t figured out what the second part of the condition means.

2)(xx xy

xx yy yxyx yy

f ff f f

f f

The matrix above is called the Hessian.

http://www.math.harvard.edu/archive/21a_fall_08/exhibits/fxy/index.html

The link above contains an animated illustration of the mixed partial derivatives. The next page contains a

proof of the theorem above. It is clear that xxf measures the concavity of the surface in the x-direction,

and yyfmeasures the concavity in the y-direction. You may be thinking, why not just check saddle points

by taking the product of the second order partials in those directions. Clearly, if the product is negative, the concavity is different in those directions, and if the product is positive, the concavity in the x and y directions are the same. However, it is possible that the concavity changes in a different direction.



Proof: The concavity in a direction u is given by

2

1

2

2

1 2

1

( )

( ) by definition of derivative of in the direction

( ( ))

( ( by linearity of derivative

=( (

of

by

)

definition

)

· · ) of de)

y

x y

x

x y

D f D

f u f u f u

D u

D f

D

f f

f f

D u

u u u u

u u u

u

u u

1 2 1 1 2 2

2 21 1 2 2

)

rivative of partial in direction of

( (

* 2 by algebraic manipu at on

)

l i

xx xy xy yy

xx xy yy

u f u u u f u u

u

u

f f

f f u u f u

If you fix a point 0 0( , )x y , and consider the second derivative 0 0

2

( , )( )

x yfDu as you change the direction¸

u. If the sign of 0 0

2

( , )( )

x yfDu changes as u changes, then the point 0 0( , )x y is a saddle point. The

computation above, will allow us to identify the points at which the concavity may change. If the function

is differentiable with continuous second partials in a neighborhood of 0 0( , )x y , in order for the concavity

to change sign, it must pass through zero. Therefore, we should look for the zeros of 0 0

2

( , )( )

x yfDu . To do

this, think of the equation (*) as a quadratic equation in 1u .

2 21 1 2 20 2 xx xy yyf f u f uu u

By the quadratic formula,

2 2 22 2 2

1

22 2

2

2

2 4 4

2

2 2

2

xy xy xx yy

xx

xy xy xx yy

xx

xy xy xx

xx

f u f u f f

f

f u f f f

f

f f fyu

u

u

f

f

y

u

As you can see, the existence of a solution to the equation 0 0 )

2

( ,( ) 0

x yD f u depends on the discriminant

of the quadratic formula. If the discriminant is positive, there are two solutions, and the sign will change as the derivative passes through this root.

2

2

0

iff 0

xy xx yy

xx yy xy

f f f

f ff

The point

0 0( , )x y is a

saddle point. If the discriminant is negative, then there are no solutions and the sign of the second derivative cannot change sign.

2

2

0

iff 0

xy xx yy

xx yy xy

f f f

f ff

The point

0 0( , )x y is a local

extrema. If the discriminant is zero, then there is only one solution to the equation.

2 0xy xx yyf f f Test is inconclusive.

multivariable calculus 14.4 – 14. 7 name - … multivariable calculus, continuous functions...

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