I'm trying to get a better feel for the geometric interpretation of the gradient of a function. Assuming you have a function in the 3 dimensional Cartesian coordinate system. I'm having a problem seeing these three geometric facts of the gradient.
1.) The gradient is a vector at each point along the surface that points in the direction of the maximum rate of change.
2.) The gradient is orthogonal to the level surface of the function
3.) The gradient vector is orthogonal to the surface.
I'm confused about how these three geometric properties can be all be true. If the gradient vector points in the direction of the maximum rate of change doesn't that imply that it is tangent to the plane? How can the gradient vector also then be orthogonal to the surface. Any help or links to good visuals would be greatly appreciated.