Direction for maximum rate of change of V at P(2, -1, 2)

[teng125]

potential V at the point P(2, -1, 2) in a rectangular
coordinate system is V (x, y, z) =x^2+4y^2+9z^2.


Find the direction that produces the maximum rate of change of V at P.

the max rate of change 37.094.
how to find direction that produces the maximum rate of change ??

 

[Astronuc]

Do you know the definition of the gradient.

The gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

http://en.wikipedia.org/wiki/Gradient

Perhaps a more reliable site -
http://mathworld.wolfram.com/Gradient.html

Gradient of V -> $\nabla$V

 

[HallsofIvy]

Also, if f is in x,y,z coordinates, then $\nabla f= \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}+ \frac{\partial f}{\partial z}$.