# Dot Product Involving Path of a Curve

1. Feb 13, 2013

### Karnage1993

1. The problem statement, all variables and given/known data
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 $\gamma ' (t)$

2. Relevant 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.

3. 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?

2. Feb 13, 2013

### HallsofIvy

Staff Emeritus
Note that $f(\lambda(t))$ is a constant for all t and use the chain rule.

3. Feb 13, 2013

### Karnage1993

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

Last edited: Feb 13, 2013