What is the Geometric Meaning of Grad f for Functions of Several Variables?

[wayneckm]

Hi everyone, nice to meet u all, this is my first post here.
I have thought of the geometric meaning of grad f for a long time, but i still can't understand what it means on the 3D/2D geometry.
Is it meaning the vector perpendicular to the curve at any point ?
thx all.

 

[Peter VDD]

gradient f of a surface = [ diff(f,x), diff(f,y), ... ];

so that means, the partial derivates to each variable. So the direction it changes fastest in. Thus perpendicular to the "touching plane" (dunno the correct word), so indeed it is perpendicular to the surface as well, but beware, it is directed towards the inside of a curved surface.

Hope it helps.
Greetz, Peter

 

[Gza]

gradient f of a surface = [ diff(f,x), diff(f,y), ... ];

so that means, the partial derivates to each variable. So the direction it changes fastest in. Thus perpendicular to the "touching plane" (dunno the correct word), so indeed it is perpendicular to the surface as well, but beware, it is directed towards the inside of a curved surface.

Hope it helps.
Greetz, Peter

a few of the concepts you mention probably won't be obvious to the original poster. The fact that the gradient is a vector with components being the partial derivatives of the function the gradient operates on, doesn't immediately imply that it points in the direction the function increases fastest in. To make that connection, you have to invoke the concept of the directional derivative, and one of it's mathematical forms as the dot product of the gradient with a unit vector pointing in the direction you wish to determine the rate of change for the function in question. You maximise the scalar value of a dot product of two vectors when they point in the same direction, therefore, the value of the rate of change is greatest when the gradient and the unit vector pointing in the direction you wish to determine the rate of change of both point in the same direction. In plain english, the gradient points in direction of greatest change. The proof of why it is tangent to a level surface for a function of 3 variables, and similarly tangent to a level curve for a function of 2 variables is something i can prove to you as well, but i'd probably have to draw it out with pictures and such for you to get a geometric feel for the situation (which i'd be happy to do, by scanning and posting it) sorry for the long winded reply, but i attempted to explain it verbally without using any symbols, which probably costed me some effort and clarity.

 

[wayneckm]

thx for all ur replies.
So, can u draw the geometric picture to me to show ??
thx!

 

[Jimmy Snyder]

Think of a terrain map. There are lines, called contour lines, that mark where the the height above sea level is some constant. If you walk along one of these contour lines, you will neither ascend nor descend. If you walk perpendicular to one of these lines, you will be ascending or descending at the greatest slope for that location. This is the direction of the gradient. Of course, the contour lines are all kinds of wiggly. Think of climbing an actual hill. If you wanted to always climb along the steepest direction, you would constantly adjust your direction to stay perpendicular to the particular imaginary contour line that you happened to be crossing at that moment.

The magnitude of the gradient is related to how 'crowded' the contour lines are. Where the contour lines are densely packed, it means the slope is large and so the gradient is large. Where the contour lines are widely separated, it means the slope and gradient are small.

 

[mathwonk]

the gradient is a vector that tells the rate of change of the function ina given direction by dotting with that direction. hence for a curve along which the function is constant, since the rate of change tangent to that curve is zero, the dot rpoduct must be zero.

so the gradient is eprpendicualr to a "level curve" of the function.


i.e. gradf is perpendicular to the curve f = constant, and at any point.