Find Steepest Climb of this Hyperbolic Paraboloid

1. Oct 11, 2009

UziStuNNa

Last edited: Oct 12, 2009
2. Oct 11, 2009

lanedance

if you define F(x,y,z) = x^2-y^2+z

then the paraboloid is defined by the level surface
F(x,y,z) = 0

the gradient direction represents the direction of maximum change, and will by defintion be perpindicular to any level surface.

f(x,y) = z = y^2-x^2

the calculate the 2D gradient
$$\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x})$$

this will represenet the direction of greatest change of z = f(x,y), with x & y and i think it should be easy to relate the slope to the magintude of the gradient

3. Oct 11, 2009

UziStuNNa

Thank you for the help, but one more question...
Do I need to multiply the gradient of F(x,y) by the unit vector to find the direction of greatest ascent?

4. Oct 11, 2009

lanedance

no worries, but the question you asked doesn't really make sense, as the gradient is a vector, so what do you mean multiplying a vector by a vector?

the steps i outlined, you will give you the x,y direction, which direction of greatest ascent & the value of that slope in that direction

if you want to find the unit vector representing the direction on surface, use the x,y direction with the slope to find a vector in that direction (in 3D), then normalise by divding by the vector magnitude

note as a check, the vector you find will be perpindicular to the gradient you gave in you first post