Question

If $ (x,y,z) = x2-y2+2yz+2z2 , gradient of
• at (1,2,1) is​

Answer 1

Answer:

x2-y2+2yz+2z2

Step-by-step explanation:

pls don't follow me

Answer 2

Answer:

The gradient of the function $f(x,y,z)=x^{2} -y^{2} +2yz+2z^{2}$ at a point  $(1,2,1)$ is $\nabla f =2i+8k$

Step-by-step explanation:

The del operator is an operator used in vector calculus as a vector differential operator

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

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

$\nabla f =\frac{\partial f }{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k$

from the question, it is given that the function is

$f(x,y,z)=x^{2} -y^{2} +2yz+2z^{2}$

the gradient $\nabla f=\frac{\partial \left(x^2-y^2+2yz+2z^2\right)}{\partial x}i+\frac{\partial (x^{2} -y^{2} +2yz+2z^{2}) }{\partial y}j+\frac{\partial (x^{2}- y^{2}+2yz+2z^{2} )}{\partial z}k$

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

$\frac{\partial f}{\partial x}=2x$

$\frac{\partial f}{\partial y}=-2y+2y=0$

$\frac{\partial f}{\partial z}=2y+4z$

now the given point is (x, y, z)=(1,2,1)

by substituting the values we get the gradient is $\nabla f =2i+8k$