October 11th, 2017, 04:43 PM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 195 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: 578 Thanks: 345 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 codimension 1, then computing a normal vector is equivalent to computing $\nabla f$ which is what you have done. Last edited by SDK; October 11th, 2017 at 07:09 PM. 

Tags 
gradient, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Question about Gradient, Directional Derivative and normal line  zollen  Calculus  10  January 30th, 2017 02:44 PM 
Gradient Question  Tzad  Calculus  1  July 14th, 2015 01:18 PM 
gradient of y= 2x +8 ??  coachcft  Calculus  2  December 15th, 2012 12:25 PM 
Gradient  truthseeker  Linear Algebra  12  October 2nd, 2012 07:07 AM 
Gradient  veronicak5678  Real Analysis  0  May 6th, 2012 09:08 AM 