Linear Algebra Done Right · Problem 6A.33

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose

f, g

are differentiable functions from

\mathbf{R}

\mathbf{R}^n

. ((a)) Show that

\langle f(t), g(t) \rangle' = \langle f'(t), g(t) \rangle + \langle f(t), g'(t) \rangle.

((b)) Suppose

c

is a positive number and

\|f(t)\| = c

for every

t \in \mathbf{R}

. Show that

\langle f'(t), f(t) \rangle = 0

for every

t \in \mathbf{R}

. ((c)) Interpret the result in (b) geometrically in terms of the tangent vector to a curve lying on a sphere in

\mathbf{R}^n

centered at the origin. A function

f: \mathbf{R} \to \mathbf{R}^n

is called differentiable if there exist differentiable functions

f_1, \ldots, f_n

from

\mathbf{R}

\mathbf{R}

such that

f(t) = (f_1(t), \ldots, f_n(t))

for each

t \in \mathbf{R}

. Furthermore, for each

t \in \mathbf{R}

, the derivative

f'(t) \in \mathbf{R}^n

is defined by

f'(t) = (f_1'(t), \ldots, f_n'(t))