How Do You Find the Gradient Vector from a Directional Derivative?

[ilyas.h]

Homework Statement

$$D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u$$

$$D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2}$$

where $$u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2})$$

find $$\bigtriangledown f(a.b)$$

Homework Equations

The Attempt at a Solution

first you change grad f into it's partial derivative form and then take the dot product:

(df/dx, df/dy).(1/root2, 1/root2) = 3root2

you'll find that:

df/dx + df/dy = 6

where would I go from here? quite confused.

 

[Ray Vickson]

Homework Statement

$$D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u$$

$$D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2}$$

where $$u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2})$$

find $$\bigtriangledown f(a.b)$$

Homework Equations

The Attempt at a Solution

first you change grad f into it's partial derivative form and then take the dot product:

(df/dx, df/dy).(1/root2, 1/root2) = 3root2

you'll find that:

df/dx + df/dy = 6

where would I go from here? quite confused.

There are infinitely many different gradient vectors that satisfy the given condition. You need more conditions in order to get a unique answer.

 

[ilyas.h]

There are infinitely many different gradient vectors that satisfy the given condition. You need more conditions in order to get a unique answer.

so is my solution complete? if not, where would I obtain more equations to make the conditions more robust?

you could say that:

grad f(6-y, y) . (1/root2, 1/root2) = 3root2

 

[Ray Vickson]

so is my solution complete? if not, where would I obtain more equations to make the conditions more robust?

you could say that:

grad f(6-b, b) . (1/root2, 1/root2) = 3root2

You cannot pull more information out of the air; somebody has to give it to you. If they do not give you more information, you have gone as far as you can go.

 

[ilyas.h]

You cannot pull more information out of the air; somebody has to give it to you. If they do not give you more information, you have gone as far as you can go.

great, so df/dx + df/dy = 6 is the final solution?

sorry for bugging you, this question is quite important to me.

 

[Ray Vickson]

great, so df/dx + df/dy = 6 is the final solution?

sorry for bugging you, this question is quite important to me.

Asked and answered.