Calculus Calculus Math Forum

 October 11th, 2017, 04:43 PM #1 Senior Member   Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics Gradient question Hello forum, A couple friends and I have been discussing a problem. Consider a field $\phi(\vec{r}) = x^2 + sin (y) - xz$ I want to find a unit vector normal to the surface $\phi(\vec{r}) = 5$ at the point (x,y,z) = (1, $\frac{\pi}{2}$, -3) My plan of attack was to $\nabla \phi(\vec{r})$ at the points (1, $\frac{\pi}{2}$, -3). = $<2,0,0>$ After I normalized the vector I got $<1,0,0>$ Then we realized we never utilized the information $\phi(\vec{r}) = 5$ given in the problem. Where did I go wrong? Thanks! October 11th, 2017, 07:07 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 621 Thanks: 392 Math Focus: Dynamical systems, analytic function theory, numerics You did use the information whether or not you realize it. Its a simple fact that if one $f$ is a smooth funcion, then $f(x) = c$ is locally a graph as long as $x$ is a regular point (i.e. as long as $\nabla f(x) \neq 0$. Now, let $M = \{x : f(x) = c\}$ and let $\gamma(s)$ be a parameterized curve lying in $M$. Then we have $f(\gamma(s)) = c$ so by differentiating we obtain $\nabla f(\gamma(s)) \cdot \frac{d}{ds}\gamma(s) = 0.$ Since $\frac{d}{ds}\gamma(s)$ must be a vector in the tangent space, $TM_{\gamma(s)}$. Since $\gamma$ was an arbitrary curve, one immediately recovers that if $x \in M$, then for any vector $v \in TM_x$ we have $\nabla f(x) \cdot v = 0$. Or to put it in English, the gradient vector at a regular point is always orthogonal to the tangent space of the level surface (if the surface is sufficiently smooth). When this tangent space has co-dimension 1, then computing a normal vector is equivalent to computing $\nabla f$ which is what you have done. Thanks from topsquark Last edited by SDK; October 11th, 2017 at 07:09 PM. Tags gradient, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zollen Calculus 10 January 30th, 2017 02:44 PM Tzad Calculus 1 July 14th, 2015 01:18 PM coachcft Calculus 2 December 15th, 2012 12:25 PM truthseeker Linear Algebra 12 October 2nd, 2012 07:07 AM veronicak5678 Real Analysis 0 May 6th, 2012 09:08 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top      