Partial Derivatives - OoCities



Partial Derivatives

Functions of several variables

• Definition

[pic]maps each ordered pair[pic]to a unique number [pic].

[pic]maps each ordered triple[pic]to a unique number[pic].

• Level curves

The level curves of [pic]have equations [pic].

[pic]describes a surface above and/or below the x-y plane.

If the surface is cut by the horizontal plane [pic]and the resulting curve

is projected onto the x-y plane we obtain the level curve of height k.

• Level surfaces

The level surfaces of [pic]have equations [pic].

If [pic] assigned a temperature to each point in space, then the

level surfaces would be surfaces of constant temperature.

Limits and continuity

• Path dependent limits

For functions of two variables there are an infinite number of curves along which the

point [pic] may approach the point [pic]. Once we choose a particular path we

may evaluate the limit using single variable methods.

If C is a smooth curve with parametric representation [pic]and

[pic] then [pic].

• Definition of a limit

[pic] means that z approaches L as the point [pic]

approaches [pic] independent of path.

• Continuity

A function of two variables is continuous if it represents a surface without any holes,

tears or gaps. Small changes in the independent variable must result in small changes

in the dependent variable.

A function[pic] is continuous at the point[pic]if

i) [pic] is defined

ii) [pic] exists

iii) [pic]

Partial derivatives

• First partials

[pic]

• Second partials

[pic]

Tangent planes

The equation of the tangent plane to the surface described by [pic] at the point

[pic] is given by

[pic]

For the special case of [pic]the equation simplifies to

[pic]

Differentiability

• Increments

[pic]is called the increment of f and is the actual change

in the function f as [pic] is moved to [pic].

• Total differentials

[pic]is called the total differential of f and is the tangent plane

approximation to the change in f as [pic] is moved to [pic]. The

total differential is actually the equation for the tangent plane in local coordinates

centered at the point of tangency.

• Differentiability

A function is differentiable if it possess the property of local linearity. For functions

of two variables this means we can accurately approximate the surface using a tangent

plane.

Definition: A function [pic] is differentiable at the point [pic] if

[pic]and [pic] exist and [pic]can be written as

[pic]

where[pic] as [pic].

Theorem: If [pic] has continuous first partial derivatives in a neighborhood of

[pic] then [pic] is differentiable at the point [pic].

Theorem: Differentiability [pic] continuity.

Chain rule

• One independent variable[pic] with [pic].

[pic]

• Two independent variables [pic] with [pic].

[pic]

Directional derivatives

• Gradient

The gradient maps a scalar field to a vector field.

Two variables [pic]

Three variables [pic]

• Directional derivatives

The derivative of [pic]in the direction of the unit vector [pic]

is computed using [pic].

• Maximum increase/decrease theorem

[pic] points in the direction of the maximum rate of increase of [pic]and

[pic] is the maximum rate of increase of [pic].

[pic] points in the direction of the maximum rate of decrease of [pic]and

[pic] is the maximum rate of decrease of [pic].

• Level curves and surfaces

[pic]is orthogonal to the level curve of [pic]passing through the

point [pic].

[pic]is orthogonal to the level surface of[pic]passing through the

point [pic].

Extreme values

• Global extrema

Theorem[Extreme value] If [pic]is continuous on a closed and bounded set D

then,[pic] has both a global minimum and a global maximum on D.

Definition[Critical point] A point [pic]in the domain of the function [pic]is

called a critical point if

xvi) [pic]

xvii) [pic]is undefined

xviii) [pic]is a boundary point

Theorem[Candidates] The extreme values of a function can only occur at a critical

point, they cannot occur anywhere else.

Extreme values [pic] critical point

Critical point [pic]extreme value

• Local extrema

It is possible to test smooth critical points to see if they are local maximums or

minimums using a two dimensional version of the second derivative test.

Theorem[Second Derivative Test]

Suppose the second partial derivatives of [pic]are continuous in a neighborhood

of the point [pic] and [pic]. Define

[pic]

i) If [pic] and [pic]then [pic]is a local minimum.

ii) If [pic]and [pic]then [pic]is a local maximum.

iii) If [pic] then [pic]is a saddle point.

iv) If [pic]then the test is inconclusive.

Lagrange multipliers

To maximize or minimize [pic]subject to the constraint [pic]

i) Solve the system [pic].

ii) Evaluate[pic]at each point obtained in step (i) and choose the maximum

or minimum value.

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