How to find the maximum vector in a vector field?

Question:

How to find the maximum vector in a vector field?

The maximum growth:

We can find the maximum increment of a function in the vector field using the modulus of its gradient. Furthermore, with the gradient we obtain the direction of the maximum increase of the function.

The maximum growth is:

{eq}| \nabla f(x,y,z) | \\ {/eq}

we can find the direction of maximum growth of the function found the gradient: {eq}\nabla f(x,y,z) \\ {/eq}

and the maximum growth can be found by finding the modulus of the gradient {eq}| \nabla f(x,y,z) | \\ {/eq}

Example:

We have,

{eq}f(x, y, z)= x^2+y^2+z^2 {/eq}

The direction of maximum growth of the function is:

{eq}\nabla f(x,y,z) = (\frac{d}{dx} ) i + ( \frac{d}{dy}) j + (\frac{d}{dz}) k {/eq}

Then,

{eq}\frac{d}{dx} ( x^2+y^2+z^2 ) = 2x \\ \frac{d}{dy} ( x^2+y^2+z^2 ) = 2y \\ \frac{d}{dz} ( x^2+y^2+z^2 ) = 2z \\ {/eq}

The maximum growth is:

{eq}| \nabla f(x,y,z) | \\ = \sqrt {{2x}^{2}+{2y}^{2}+{2z}^{2}} \\ =2\,\sqrt {{x}^{2}+{y}^{2}+{z}^{2}} \\ {/eq}