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Introduction
Text Calculus is the study of functions.
In the study of functions of
one variable, we encounter domains and ranges of functions, function
graphs, and properties of functions such as continuity. In multivariable
calculus we will extend all of these notions to functions of two and
eventually more variables. We will also deal with functions of one
variable associated with functions of a single variable, namely slice
curves and contour lines.
A point is given in polar coordinates by an angle θ about the origin and a real number r, the distance from the origin.
It is customary to write this as [r,θ].
A point can be converted from polar to Cartesian coordinates using the formulas
\[ x = r \cos(\theta), \]
\[ y = r \sin(\theta). \]
Similarly, a point can be converted from Cartesian to polar coordinates using the formulas
\[ r = \sqrt{x^2 + y^2}, \]
\[ \theta = \arctan (y/x) . \]
Examples Zero Functions
The simplest function of all is the zero function, defined by f(r,θ) = 0 for all r,θ. This function can be defined for any domain, and the range will always always be the single point { 0 }.
Constant Functions
The next simplest class of functions are the constant functions
defined by f(r,θ) = k for all r,θ. A constant function can be defined for any domain, and the range will always always be the single point { k }.
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