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### Inner Products, Gradients, and Differentials

In flat spaces, we often identify the differential of a function with its gradient. However, when dealing with a more general setting, one can run into problems making sense out of such a definition.

For example, the gradient is a vector, and it should be possible to think of vectors as infinitesimal displacements of points. In , any infinitesimal displacement must satisfy . Thus, may not always be a vector, since it does not necessarily satisfy this equation. A gradient should be an infinitesimal displacement that points in the direction of the displacement which will give the greatest increase in .

If the tangent space has an inner product, though, one can find a useful way to identify the uniquely with a tangent vector. Let be a symmetric nondegenerate bilinear form on the tangent space of at . Then one can define the gradient, , implicitly by

Since is a nondegenerate form, this is sufficient to define . The function tangent carries out this projection from differentials to tangents (shown in Figure 9.7). This operation is performed by grad to produce the gradient of the objective function.

Next: Getting Around Up: Geometric Technicalities Previous: The Difference Between a   Contents   Index
Susan Blackford 2000-11-20