I am in calculus 3 and I have a question on gradient versus tangent. I know that this has been answered several times before:
But I still have much trouble understanding them.
Assume $z = x^2 + y^2$ (a sphere)
In the topic of directional derivatives and gradients, gradient is the steepest slope (or, more formally, the direction to move so that z increases most rapidly). AT THE SAME TIME, it is a normal to the level curve (to my understanding, it means when z is fixed at a particular value, a vector that is perpendicular to the tangent of a curve at a particular point. The gradient points to the concave side of the level curve).
Combining the two ideas above, I assume it is correct that a normal to the level curve is the same as the direction in which z will increase most rapidly (although I am unsure why it is true).
Somehow, then I thought that gradient points to the direction where a tangent line to the 3D object is, because gradient points to the direction of the steepest slope.
And then in the following topic of tangent planes, now it tells me that gradient is actually a normal line instead of a tangent line to a 3D object? What a surprise!
In the case of 2D, gradient is the slope of a line. therefore, I naturally think that in the case of 3D, gradient is also the slope (or, the vector that points to the steepest slope for z). It is not the case? What is wrong with my thinking?