Dao asked in Science & MathematicsMathematics · 1 month ago

# Find the directional derivative of the function f(x,y,z)=𝑧^2 -x𝑦^2 at the point P(2,1,3)  in the direction of the vector v = < -1, 2, 2 >.?

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• 1 month ago

f(x, y, z) = z² – xy²

grad(f(x, y, z)) = <∂f/∂x, ∂f/∂y, ∂f/∂z>

= <-y², -2xy, 2z>

At the point P(2 ,1, 3) the gradient vector, g, is:

g = grad(f(2, 1, 3)) = <-1², -2*2*1, 2*3> = <-1, -4, 6>

The unit vector in the direction v=<-1, 2, 2> is:

u = v/||v|| = <-1, 2, 2>/√((-1)² + 2² + 2²) = <1, 2, 2>/√9

= ±(1/3)<-1, 2, 2>

I will use the positive (+) solution.

Now find the dot product of g and u:

g•u = <-1, -4, 6>•(1/3)<-1, 2, 2>

= [(-1)*(-1) + (-4)2 + 6*2]/3

= (1 - 8 + 12)/3 = 5/3

• Ian H
Lv 7
1 month ago

Explanation of general concept for a hill drawn on a map in contours.

If we imagine standing at a particular point on the side of a hill,

facing along a level curve, (a contour), the slope would be zero.

The maximum gradient, grad f(x, y) at that point would be vector perpendicular to the level curve and this is like the steepest slope downhill.

At the same point, but facing in any other direction, (specified by a UNIT vector), the slope will be some component of grad f(x, y), calculated as the dot product of the two vectors. That is known as the directional derivative D.

f(x, y ,z) = z^2 – xy^2

We can apply the same method with these 3 variables

grad[f(x, y, z)] = <∂f/∂x, ∂f/∂y, ∂f/∂z>

grad f = <-y^2, -2xy, 2z> ....... and this depends on position

At the particular point P(2 ,1, 3) the substituted values give

grad[f(2, 1, 3)] = <-1², -2*2*1, 2*3>

grad f at P is <-1, -4, 6>

We now need to calculate the UNIT vector u associated with v <-1, 2, 2>

√(1^2 + 2^2 + 2^2) = √(9) = 3, so,

u = (1/3)<-1, 2, 2>

Next recall that the result of calculating the dot product <a, b, c>•<p, q, r>

is the scalar ap + bq + cr

D = grad[f(2, 1, 3)] • u = (1/3)<-1, -4, 6><-1, 2, 2>

D = (1/3)(1 - 8 + 12) = 5/3

Steve4Physics found this first.

• rotchm
Lv 7
1 month ago

Just apply the formula you saw for this.

∇f • u, where u is the unit vector along your given direction; u = v/|v|.

Or, did you see a different formula for this?

Hopefully no one will spoil you the answer thereby depriving you from your personal enhancement; that would be very inconsiderate of them.

• Vaman
Lv 7
1 month ago