Partial derivatives and the calculus of surfaces
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Two-dimensional surfaces embedded in three-dimensional space
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Let us consider the case of a function of two variables, such as [Equation goes here - download the original to see it.] The value of z depends on the value of x and y. We may picture this in a 3-dimensional picture as follows [Diagram goes here - download the original to see it.] The graph as a whole forms a surface; for example, it might look like this [Diagram goes here - download the original to see it.] The surface is a two-dimensional object embedded in three-dimensional space. In order to develop a calculus of two (and multi-dimensional functions) let us recall that a derivative of a function in one-dimension finds the gradient of the tangent to the graph of that function at a given point. An alternative way of putting this is that it finds the rate of change of the function. For example, suppose we are given the function [Equation goes here - download the original to see it.] Then, by fixing a point we define a curve along the surface in the x-direction. [Equation goes here - download the original to see it.] This expression states that the function [Equation goes here - download the original to see it.] is obtained from the function [Equation goes here - download the original to see it.] by evaluating that function at the points [Equation goes here - download the original to see it.]. For example, suppose [Equation goes here - download the original to see it.] Differentiating the function finds the gradient function. Substituting a specific value of x, then finds the gradient of the tangent at that point. In general, this is [Equation goes here - download the original to see it.] For our specific example, [Equation goes here - download the original to see it.] We can illustrate these general ideas by the following diagram [Diagram goes here - download the original to see it.] The function, [Equation goes here - download the original to see it.] defines a curve along the surface running "parallel" to the x-axis (its perpendicular projected picture to the plane x, y is parallel line to the axis x), and the derivative of this function defines the gradient of the tangent along this curve (you can see it as arrows in the figure). [Diagram goes here - download the original to see it.] Of course, these ideas can be repeated for the y-direction. That is, suppose we fix a point, , and define a curve running along the surface parallel to the y-axis. [Equation goes here - download the original to see it.] Then, by differentiating this function, and evaluating it at the point, [Equation goes here - download the original to see it.], we obtain the gradient of the tangent to this curve at the point [Equation goes here - download the original to see it.] in the y-direction. [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] We can, in fact, use these ideas to obtain a sketch of the entire surface represented by the function [Equation goes here - download the original to see it.] We were considering, for example, the function [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Equation goes here - download the original to see it.] We can repeat this process for curves in the y-direction [Equation goes here - download the original to see it.] This defines a saddle surface [Diagram goes here - download the original to see it.] The curves "parallel" to the x-axis are parabolas lying on the surface. [Diagram goes here - download the original to see it.] And the curves "parallel" to the y-axis are straight lines, also lying on the surface. [Diagram goes here - download the original to see it.] However, one thing we are assuming here is that the surface does not do anything peculiar, like form little "wrinkles" To visualise what we mean by this, consider a small part of this saddle surface. [Diagram goes here - download the original to see it.] To say that this surface was "wrinkled" would mean that this disc, for example, would have folds in it. [Diagram goes here - download the original to see it.] However, this is not the case for our function [Equation goes here - download the original to see it.] Our function is polynomial, and all of polynomial functions are continuous, and differentiable. A very small part of their surfaces is similar to a plane. The surface does not present unexpected "surprises". To explain why, let us now introduce the idea of a partial derivative. The derivative of a function [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] To illustrate these ideas with regard to our example function [Equation goes here - download the original to see it.] To form the partial derivative of a function with respect to x, you simply treat the y term as a constant and differentiate. That means, for example that 3xy differentiates to just 3y. To form the partial derivative of a function with respect to y, you treat x as a constant and differentiate with respect to y. In the above example, this means that x2, since it is not a function of y, it differentiates to 0. The partial derivatives give the gradient functions of [Equation goes here - download the original to see it.] in the x-direction and y-direction respectively. [Diagram goes here - download the original to see it.] For example, for our function, the gradients of the tangents to the surface at the point [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] These partial derivatives provide us with the gradients of the tangents in two directions - the direction of the x-axis and the direction of the y-axis respectively. However, what we would also like to know is what the gradient of the tangent is in other directions.
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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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