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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