Question

find the gradient of
f(x,y,z)=x²y³z⁴​

Answer 1

Answer:

The gradient of the function  f(x,y,z)=x²y³z⁴​ is $=i(2xy^{3} z^{4} )+j(3yx^{2} z^{4} )+k(4zx^{2} y^{3} )$

Explanation:

Del operator is used in vector calculus as a vector differential operator

To change a scalar quantity to a vector quantity del operator can be used as a gradient

For the cartesian coordinate system, the gradient is given by the formula

$=id/dx+jd/dy+kd/dz$

From the question, it is given that the function is

$f(x,y,z)=x^{2} y^{3} z^{4}$

the gradient of the function is

$=(id/dx+jd/dy+kd/dz)x^{2} y^{3} z^{4}$

$=i\frac{d(x^{2} y^{3} z^{4})}{dx}+j\frac{d(x^{2} y^{3} z^{4})}{dy}+k \frac{d(x^{2} y^{3} z^{4})}{dz}$

differentiate the function with respect to x, y, and z as follows

$=i(2xy^{3} z^{4} )+j(3yx^{2} z^{4} )+k(4zx^{2} y^{3} )$

which is a vector