Question

Suppose a tub has the shape of an elliptical paraboloid given by z = ax2+by2 (where a, b are some positive constants). If a marble were released at the point (1, 1, a + b) on the inside surface of the tub, in what direction would it begin to roll? Your answer should be a unit vector in the requested direction.

Answer 1

It would roll in this direction.

`\\u = (-a/√(a^2+b^2),-b/√(a^2+b^2))`

Explanation:

It would roll to the direction of maximum decrease, which is the -1 times the direction of maximum increase, which is given by the gradient of the function.

Since

`z = ax^2 + by^2`

For this case, the gradient of your function would be

`\\abla z = (2ax , 2by)`

And -1 times the gradient of your function would be

`-\\abla z = (-2ax , -2by)\\`

Then, at

`(1,1,a+b),\\x = 1 \\y = 1`

So it would go towards

`v = (-2a,-2b)`

The magnitud of that vector is

`|v| = 2√(a^2+b^2)`

and to conclude it would roll in this direction.

`\\u = (-a/√(a^2+b^2),-b/√(a^2+b^2))`