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Unformatted text preview: ary motion based
on astronomical observation.
i.
ii.
iii. Sun it lies on the focus of the path of the planet, the planet travels in an ellipse
Line Sun planet sweeps out constant area through constant time
Period of Revolution Isaac Newton derived mathematically these laws as consequences of the second law of motion: 14 Chapter 14: Partial Derivatives 15 14.1 Functions of Several Values
With a one variable function, there is one input to one output, however for an n variable
function; there are n inputs but only one output.
Definition (Function of n Variables): An assignment of a real value, Z, to any point in a set The set D is called the domain of the function.
If the function is f then: We often write:
Independent variables:
Dependent variable: Z
Note: A function of n variables may be considered as: A function of n inputs
A function of a vector with n coordinates:
As a function of a point: Definition (Linear Function): Any function of the form: Consider a two variable function
Definition (Level or Contour Curves): The curves
called level or contour curves of f. 16 ,where k is a constant, are 14.2 Limits and Continuity
Recall (1 Variable Limit):
means that such that Definition (Limit):
that , suppose there exists a disk U around a point Then means such such that: Independence of Path
The definition of the limit takes care of the fact that if
approaches L along any path that approaches then Nonexistence of a Limit
If
path as
with along a path
then and
does not exist. as along a Note: All limit laws and squeeze theorem hold true for both single variable limits and
multivariable limits
Definition (Continuity):
we say is called continuous on a point
if
is continuous on if it is continuous on every point . Note: All continuous laws hold for both single variable and multivariable limits.
i.e.: sums/product/etc of continuous functions is also continuous
Polynomial function of two variables is the sum of terms of the form: Where
Rational function of two variables is the ratio of two polynomial functions in
All rational functions are continuous under the domain
Functions of 3 (or more) Variables
For limits and continuity, all of the generalizations in single variables work in multivariable 17 14.3 Partial Derivatives
Recall (1 Variable Derivative): Interpretation: Slope of the line tangent to at a point Two variables: Derivative??
Definition (Partial Derivative): , That is,
where
where
We may consider as functions Other notation: For second and higher order derivatives: 18 or the rate of change of at Rules: To compute
function of we consider Clairaut’s Theorem:
continuous on then: to be a constant in a disk in that is to consider . If This can be generalized for functions of 3 or more variables 19 and as a are both 14.4 Tangent Planes and Linear Approximation
Tangent Planes
: the curve of intersection of the surface with the plane : the curve of intersection of the surface with the plane : the tangent vector to at The tangent plane to S at P is the plane determined by and
approximates most closely the surface S near the point P.
In order to find the equation of the tangent plane we need
The plane will be of the form: We have: We then parameterize: We then evaluate at to get : Sim...
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 Spring '08
 Dr.AnthonyDixon

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