Answer to Question #164580 in Field Theory for faisal riaz

Question

Find the gradient vector and its modulus of scalar function (x,y,z) = x⁶y⁴z⁰-z³yx³-89 at (3,-2,1)

Answer 1

Answer

Function

V=$x^6y^4z^0-z^3yx^3-89$

So

Gradient vector

$\nabla V=(i\frac{dV}{dx}+j\frac{dV}{dy}+k\frac{dV}{dz})$

Putting value of function

$\nabla V= i(6x^5y^4-3x^2yz^3) +j(4x^6y^3-z^3x^3) +k(-3z^2yx^3)$

Now at point (3,-2,1)

$\nabla V=23382i+23355j+162k$

Now it's magnitude

$|\nabla V|=\sqrt{(23382) ^2+(23355) ^2+(162) ^2) }\\=33048.5$