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
tangentcarries out this projection from differentials to tangents (shown in Figure 9.7). This operation is performed by
gradto produce the gradient of the objective function.