Hard time visualizing gradient vector vs. tangent vector.

[moe darklight]

OK, this is really confusing me. Mostly because i suck at spatial stuff.

If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane, wouldn't it be perpendicular to the function too?

This is the Wikipedia image for the gradient of a function, and it's pretty much what I imagine when I think of it: http://upload.wikimedia.org/wikipedia/en/3/31/Gradient99.png"

but if those lines were perpendicular to the tangent planes at their given points, wouldn't they all be pointing away from the graph (like "hairs")?

Or is it just saying that it's perpendicular to the level curves? -- the pictures are very confusing and they always look like it's pointing away from a tangent plane, which makes no sense to me.

EDIT: I think I got confused because it was an example with a function of three variables f(x,y,z) and I was thinking about it in f(x,y).

 

[kzhu]

The graph at Wikipedia is a function of two variables f(x,y). If you take the gradient of this function (or a scalar field), it would be the 2-D vector as plotted on graph in the X-Y plane. The discussion of being perpendicular is in the X-Y plane but not the X-Y-Z space.

So what is tangential is the line tangential to the level curves and the gradient vector is perpendicular to these tangential lines.

If you have a function f(x,y,z), then its gradient vector would be 3-D and its level curves would be surfaces. I use to image a cloud of colored dots, whose color corresponds to f. This gradient vector would be perpendicular to the surface.

I had confused myself over this and this is my understanding

 

[richardlhp]

I think that it is not the tangent plane but the level curve that the gradient is perpendicular to, i.e. when the function is at a certain value. For e.g. a function of x and y then the level surface is not that tangent to the function, but an arbitrary curve formed by the function on any value of the function i.e. f(x,y)=c. So let us imagine a cone as an example. At any value of the function, the level curve would be like a circle on a plane parallel to the xy plane. The gradient is perpendicular to this level curve. To further determine the direction of the gradient, use the theorem that the gradient points in the direction of the highest increase.

I was also confused at first at these concepts, but after clarifying it I finally understood.

Hope this helps cheers!

 

[richardlhp]

You should think of the simplest case where f(x,y) before moving on to f(x,y,z...).