Dot Product Involving Path of a Curve

[Karnage1993]

Homework Statement

Let ##\gamma(t)## be a path describing a level curve of ##f : \mathbb{R}^2 \to \mathbb{R}##. Show, for all ##t##, that ##( \nabla f ) (\gamma(t))## is orthogonal to ##\gamma ' (t)##

Homework Equations

##\gamma(t) = ((x(t), y(t))##
##\gamma ' (t) = F(\gamma(t))##
##F = \nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}\right)## [this is called a gradient field]

None of these were given for the question at hand but I think they might be useful.

The Attempt at a Solution

If ##x(t)## and ##y(t)## are the parameters for ##\gamma(t)##, then ##( \nabla f ) (\gamma(t)) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)##

But both ##\nabla f (\gamma(t))## and ##\gamma ' (t)## are equal to ##F(\gamma(t))##, so the dot product is

##F(\gamma(t)) \cdot F(\gamma(t)) = ||F(\gamma(t))||^2##

At this point, I'm stuck. I don't think ##||F(\gamma(t))||^2## would be 0 for any ##t## let alone for any function ##f : \mathbb{R}^2 \to \mathbb{R}##. Did I make a bad assumption/simplification somewhere?

 

[HallsofIvy]

Note that [itex]f(\lambda(t))[/itex] is a constant for all t and use the chain rule.

 

[Karnage1993]

EDIT: I see now, thank you for the help!