Answer to Question #164580 in Field Theory for faisal riaz

Question #164580

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


1
Expert's answer
2021-02-19T06:08:46-0500

Answer

Function

V=x6y4z0z3yx389x^6y^4z^0-z^3yx^3-89

So

Gradient vector

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

Putting value of function

V=i(6x5y43x2yz3)+j(4x6y3z3x3)+k(3z2yx3)\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)

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

Now it's magnitude

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



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