I've been reading the whole day and now I'm more confused then when I started. Some say that gradient is always perpendicular, and others say it always shows the direction of steepest slope, which should mean tangent? Lets take simple functions like:
$f_1(x)=x^2$
$f_2(x, y)=x^2+0.5y^2$
The gradients should be?
$grad f_1(x) = \frac{df}{dx}\hat i=2x\hat i $
$grad f_2(x,y) = \frac{\delta f}{\delta x}\hat i + \frac{\delta f}{\delta y}\hat j =2x\hat i+y\hat j $
As I get it, these vectors really point in the direction on the steepest slope, and their magnitudes are proportional to the slope steepness. But they are neither tangent or orthogonal to the curve f(x) and surface f(x,y). For the first function it is a vector on an x axis, and for the second function it a vector on an x-y plane.
But if I write the functions differently:
$x^2-y=0$
$x^2+0.5y^2-z=0$
And then do partial derivatives of the left sides, I get some components:
$ 2x\hat i -\hat j $
$2x\hat i+y\hat j-\hat k$
These vectors seems to be orthogonal to the curve and the surface, but what are they?
Also, what would be the gradient of a plane that is angled 45° to the x-y plane cutting it along x axis? Where would it point to, and what would be it's vector components.