Example 3

The euclidean scalar product $(\mathbf{x}| \mathbf{y})$ (Gen. Top , VI, p. 40) is a bilinear map (necessarily continuous) of $\mathbf{R}^n \times \mathbf{R}^n$ into $\mathbf{R}$. If $\mathbf{f}$ and $\mathbf{g}$ are two vector functions with values in $\mathbf{R}^n$, and differentiable at the point $x_0$, then the real function $x \mapsto (\mathbf{f}(x)|\mathbf{g}(x))$ has a derivative equal to $(\mathbf{f}'(x_0)|\mathbf{g}(x_0)) + (\mathbf{f}(x_0)|\mathbf{g}'(x_0))$ at the point $x_0$. There is an analogous result for the hermitian scalar product on $\mathbf{C}^n$, this space being considered as a vector space over $\mathbf{R}$.

Let us consider in particular the case where the euclidean norm $\| \mathbf{f}(x) \|$ is constant, so that $(\mathbf{f}(x)|\mathbf{f}(x)) = \| \mathbf{f}(x) \|^2$ is also constant; on writing that the derivative of $(\mathbf{f}(x)|\mathbf{f}(x))$ vanishes at $x_0$ we obtain $(\mathbf{f}(x_0)|\mathbf{f}'(x_0)) = 0$; in other words, $\mathbf{f}'(x_0)$ is orthogonal to $\mathbf{f}(x_0)$.