Partial derivatives and the calculus of surfaces
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Multiple variable calculus. Functions of more than one variable
Equations are omitted for technical reasons - download the original pdf
The equation [Equation goes here - download the original to see it.] is one-dimensional. The dependent variable, y, depends on how just one other independent variable, x, changes. This is fine for describing a great number of physical situations in the world. For example, the equation [Equation goes here - download the original to see it.] may describe the population of a country as it grows exponentially over time. [Diagram goes here - download the original to see it.] As another example, the equation [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] describes harmonic oscillation of a particle. Both of these functions arise as the solutions of one-dimensional differential equations. However, it is quite clear that this one-dimensional case is only of limited application, and that in many other situations we expect the dependent variable to be determined by two or more independent variables. In the two-dimensional case we write the function like this [Equation goes here - download the original to see it.] That is, z is a function of two dependent variables, denoted here by x and y. Of course, there is no particular commitment to the use of the symbols z, x and y to denote the variables. For example, [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] also represent two dimensional functions. When we use more than two dimensions, the use of subscripts to denote the various independent variables, can be very useful. [Equation goes here - download the original to see it.] is a three-dimensional relationship, and the general case is [Equation goes here - download the original to see it.] which denotes an n dimensional function.
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Contents of
Partial derivatives and the calculus of surfaces
1 Multiple variable calculus. Functions of more than one variable
2 Two-dimensional surfaces embedded in three-dimensional space
3 The tangent plane
4 Stationary points
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