Maximizing Scalar Increase: Understanding the Direction of the Gradient

[daudaudaudau]

Hi. The book I'm reading says "We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar". But how does one know in which direction the maximum increase is?

 

[mda]

I suggest you consult standard sources to see how the gradient is calculated. Wikipedia is usually a good start.

 

[daudaudaudau]

I know how to calculate it. Then I guess my qustion is why the gradient is pointing in the direction of maximum increase.

 

[mathboy]

gradient involves dot product, and cos is maximized when the angle is 0. That gives the direction for maximum rate.

 

[Vid]

Start from the directional derivative. grad(f)*u. This is equal to |grad(f)||u|cos(x), where x is the angle between them. This is maximum when cos(x) = 1 which occurs when x = 0. Implying that the maximum rate of change is in the direction of the gradient itself.