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Unformatted text preview: ary motion based
on astronomical observation.
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:
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 Non-existence of a Limit
with along a path
does not exist. as along a Note: All limit laws and squeeze theorem hold true for both single variable limits and
we say is called continuous on a point
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,
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
: 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