Tangent Planes
Some important formulas:
Gradient Vector or Gradient of f(x,y)
obtained by evaluating the partial derivatives of f at P.
The directional derivative of f(x,y) at P in the direction of unit vector u is
The function f increases most rapidly at P in the direction of the gradient of f at P and decreases most rapidly at P
in the direction of negative of the gradient of f at P.
For more details on these concepts go to the website:
Tangent Plane and Normal Line:
The normal line of the surface at P is the line through P and parallel to the gradient of f at P.
So the equation of the tangent plane is
The equation of the normal at P is
For more details on these concepts go to the website:
Finding the equations of a line tangent at P to the curve of intersection of surfaces f and g.
The tangent line is perpendicular the vectors u = gradient of f and v = gradient of g at P. So it is parallel to the vector given by the cross product of u and v.