Partial derivatives and the calculus of surfaces
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Stationary points
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Since we are able to define the tangent plane and its normal for any point on the surface, we are also able to find the stationary points on the surface. The stationary points will be points where the tangent plane is parallel to the x, y-plane. [Diagram goes here - download the original to see it.] At saddle points the surface both bends inwards and outwards, thus [Diagram goes here - download the original to see it.] The criterion for finding a stationary point on a surface is that the gradients of the tangents in the tangent plane at that point must be zero in any direction. This means that the partial derivatives are zero at a stationary point. For example, for a two dimensional surface [Equation goes here - download the original to see it.] This is illustrated by the following example Example: A surface has equation [Equation goes here - download the original to see it.] Solution: [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.] [Equations go here - download the original to see it.]
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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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