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Calculus for Physics 50 series-8_4

# Calculus for Physics 50 series-8_4 - Calculus for Physics...

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Calculus for Physics 50 series an 8-page guide to concepts important for these courses. By Todd Sauke We start with the important concept of a Function . A function is a mathematical relation for which, if given an input value (the "argument"), the result (the "output" value of the function for that argument) is given by exactly one number. In the same way that we can refer generally to an arbitrary or unknown quantity as "x", we can refer generally to some arbitrary or unknown function using an abstraction like "F(x)". A function can be thought of as a mathematical operation (or " rule ") that produces a single output number for any input number in the domain of the function. The specific relation, F(x) = x 2 + 1 is a function; for every value of the input, x, you can compute exactly one output value: F(3) = 3 2 + 1 = 10, for example. The input value of a function is also called an " independent variable ". f(x) = x ½ (the square root of x) is not a function, since the result, for example, for x=4 gives plus or minus 2 ( two possible results, rather than exactly one result. A function can be plotted as a graph showing a curve of any shape, as long as there is exactly one and only one value corresponding to each input value. We can also have functions with multiple arguments (or multiple independent variables), say a function of x, y, and z such that there is exactly one output result for any combination of input values x, y and z. For example, F(x, y, z) = 3 + 2xy 2 – z 4 is a function of x, y and z. For x=1, y=3 and z=2, F(1, 3, 2) = 3 + 18 – 16 = 5, exactly one output number corresponding to the specific input values. Sometimes a curve on a graph, say the circle in the x,y plane given by the equation x 2 + y 2 = R 2 , does not represent a function y(x). However, this same curve could be considered a function r( θ ) by changing from Cartesian coordinates to polar coordinates . In that case, each value of θ corresponds to exactly one output value r. Trigonometry and polar coordinates (version 3 9-03-2008)

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A vector function is basically the same as a regular (scalar) function, except that for every input value (or combination of input values, for functions of several variables) the output is exactly one vector rather than exactly one number as for scalar functions. A vector, v , has a size (magnitude) and a direction and is written as a symbol name in bold type, often with a little arrow or line above it. A three-dimensional vector has three components , corresponding to the three spatial dimensions x, y, and z; the x, y, and z components of v are written as v x , v y , and v z , respectively. A unit vector is a vector of magnitude equal to one, whose only job is to specify a direction for a vector component. The unit vectors for specifying the x, y, and z directions are written as î , ĵ , and ǩ , or sometimes as x , ŷ , and ź respectively. The vector, v , is written in component form as v = v x î + v y ĵ + v z ǩ . A vector equation is just "shorthand" for three separate equations involving the respective components.
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Calculus for Physics 50 series-8_4 - Calculus for Physics...

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