In functions involving only two variables the gradient is supposed to be the instantaneous rate of change of one variable with respect to the other and this is usually TANGENT to the curve. So then why is the gradient NORMAL to the curve at that point, since it is supposed to represent the direction of maximum increase?(adsbygoogle = window.adsbygoogle || []).push({});

Same thing for 3 dimensions. Shouldn't the direction of maximum increase be some vector tangent to the level surface at a point rather than orthogonal? Yet the gradient vector is the direction of maximum increase and is orthogonal. Also I would like to know what exactly the gradient vector represents, since I have usually understood derivatives to be of one variable with respect to another eg, "dy/dx". Than what exactly does "d/dx" or "d/dy" represent (ie. what is their physical/geometric interpretation, how can I visualize this?). And by extension, what does a gradient vector in 3d represent, an increase in exactly WHAT? I know that when the equation of a plane is given in the form:

z = x + y

The gradient vector has no z component and represents the direction in the xy plane corresponding to maximum increase in "z". But what about when an equation is given in the form

K = x + y + z

Then the gradient vector is calculated in terms of x, y and z, but what does it now represent? Since K is a constant and neither increase nor decreases?

Thanks

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# Why is the gradient vector normal to the level surface?

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