1. Gradient and directional derivative
   • The gradient of a function at a point is a vector that points in the direction of maximum increase of the function at the point the length of gradient is the directional derivative along the gradient.
   • The directional derivative of a function is the rate at which the function changes as you move in a specified direction.
   • The following we give formula of gradient $\nabla f(a,b)$ and the directional derivative $\frac{\partial f}{\partial \vec{u}}(a,b)$ along the vector $\vec{u}$: \(\nabla f(a,b)=\left(\frac{\partial f}{\partial x}(a,b), \frac{\partial f}{\partial y}(a,b)\right)\text{ and } \frac{\partial f}{\partial\vec{u}}=\nabla f(a,b)\cdot\frac{\vec{u}}{|\vec{u}|}.\)
2. The Gradient Theorem
   • The gradient of a function at a particular point is perpendicular to the level curve of the function at that point.
   • The gradient of a function at a particular point points in the direction of the maximum rate of increase of the function at that point.
   • If the level curves of a function are densely packed in a region, then the gradient of the function is larger in that region.
3. Application of the gradient theorem to find tangent space
   • The tangent space (plain) at a point $(a,b,c)$ of a surface defined by f(x,y,z) = C is \(\nabla f(a,b,c)\cdot (x-a,y-b,z-c) =0,\) i.e. \(\frac{\partial f}{\partial x}(a,b,c)(x-a) + \frac{\partial f}{\partial y}(a,b,c)(y-b) + \frac{\partial f}{\partial z}(a,b,c)(z-c)=0\)