the path of the steepest descent?

Question

Determin the path of the steepest descent along the surface z=A+x+2*y-x^2-3*(y^2) from the point (0,0,A).

Answer 1

The path of the steepest descent is the direction opposite to the gradient. The gradient of a function is the vector whose coordinates are the 1st derivatives of the function. We have, wherer d means partla derivative:

dz/dx = 1 - 2x, therefore dz/dz ((0,0) = 1
dz/dy = 2 - 6y, therefore dz/dy(0,0) = 2

So, the path from (0,0, A) is the vector v = u(-1, -2), where u is a scalar, on the XY plane,

Answer 2

Use the gradient.