Thread: equation for stream's path

Hi, i'm new.
I have a question that gives me the equation of a hill:
h(x,y)=40/(4+x^2+3y^2)
I need to find an equation for the path of a stream that follows the path of steepest descent down the hill and passes through the point (1,1,5). I'm ok with the course material (i think, hehe) - that is, i realize that i'm working with gradients, partial derivatives, etc. If this was a "find the tangent plane or line" problem i'd be ok, but it seems to be a differential equation problem. In that case, i'm not sure how to make a differential equation out of it. It'd be great if someone could help.

Thanks,
ilya

P.S. We haven't learned integrals of any sort for multivariables yet...so path integrals are out (dunno if that helps).

Originally Posted by ilya

Hi, i'm new.
I have a question that gives me the equation of a hill:
h(x,y)=40/(4+x^2+3y^2)
I need to find an equation for the path of a stream that follows the path of steepest descent down the hill and passes through the point (1,1,5). I'm ok with the course material (i think, hehe) - that is, i realize that i'm working with gradients, partial derivatives, etc. If this was a "find the tangent plane or line" problem i'd be ok, but it seems to be a differential equation problem. In that case, i'm not sure how to make a differential equation out of it. It'd be great if someone could help.

1)Find , i.e. the gradient.

2)Find , i.e. the gradiet at the point.

3)Find , i.e. the magnitude of that vector. That is the steepest point of descent.

But then you're saying that the stream flows down the hill in a straight line, the direction of which is given by (1,1)? What if the hill has a significant curvature change or "bump" of some kind, then the path of the stream would be altered as it approaches. So i guess what i'm asking is, when it says "Find the equation of the path of the stream", shouldn't it be more complicated then a line? Or maybe i'm more lost than i thought? If you could clarify, it would be appreciated.

Thanks again

Originally Posted by ilya

But then you're saying that the stream flows down the hill in a straight line, the direction of which is given by (1,1)? What if the hill has a significant curvature change or "bump" of some kind, then the path of the stream would be altered as it approaches. So i guess what i'm asking is, when it says "Find the equation of the path of the stream", shouldn't it be more complicated then a line? Or maybe i'm more lost than i thought? If you could clarify, it would be appreciated.

Thanks again

The gradient of a function always points in the direction of the steepest (positive) slope of the function at that point. (So the direction of "steepest descent" will actually be the negative of .) However given the function h(x, y, z) the steepest descent will be some kind of curve, not necessarily straight. (ie. need not be a linear function.) The function h(x, y, z) will contain any bumps you are referring to and thus will contain that information as well.

-Dan

Ok, i think i know why i'm having problems with this, so i'll just try asking a more direct question:
I have a function h(x,y) that gives me the height of a hill for a certain x and y. So in other words, i have a 3D hill that, as Dan said, contains information on the bumps and whatnot in the terrain. I can access this information for any x,y by looking at the gradient.
So now here's the problem: i don't know how to take a point (e.g. (1,1,5)) on the surface of this hill and make a continuous curve given by a function (call it f) s.t. at any height z, f will give me the x and y coordinates for the point that the stream passes through, with the condition that at any point, the stream's direction is that of the gradient of h.

So again, to clarify, i know that i can look at any point (x,y,z) on the hill and tell you where the steepest descent is, etc. But how do i take that information and make a continuous curve out of it?