Tangent Planes and Normal Vectors
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) Text If both partial derivatives of f exist at a point (x_{0},y_{0}), then the equation of the tangent plane of f at (x_{0},y_{0}) is L(x,y) = f(x_{0},y_{0}) + f_{x}(x_{0},y_{0})(x  x_{0}) + f_{y}(x_{0},y_{0})(y  y_{0}).
The general form of a plane through the point (x_{0},y_{0},z_{0}) is L(x,y) = z_{0} + p(x  x_{0}) + q(y  y_{0}) for appropriate choice of p and q. In order for the tangent lines to the slice curves at (x_{0},y_{0}) to lie in the tangent plane, we must have p = f_{x}(x_{0},y_{0}) and q = f_{y}(x_{0},y_{0}).
The tangent line to the slice curve y = y_{0} is given by z = z_{0} + f_{x}(x_{0},y_{0})(x  x_{0}). The vector (1, 0, f_{x}(x_{0},y_{0})) lies in the direction of the tangent line to the y = y_{0} slice curve.
Similarly the vector (0, 1, f_{y}(x_{0},y_{0})) is parallel to the tangent line to the x = x_{0} slice curve. The vector (f_{x}(x_{0},y_{0}),f_{y}(x_{0},y_{0}),1) is perpendicular to both of these vectors, so it lies along the normal line to the surface at the point. This normal line can be written (x_{0},y_{0},z_{0}) + u(f_{x}(x_{0},y_{0}),f_{y}(x_{0},y_{0}),1).
The unit vector in the direction of (f_{x}(x_{0},y_{0}),f_{y}(x_{0},y_{0}),1) is called the unit normal to the graph of f at (x_{0},y_{0},z_{0}), written N(x_{0},y_{0}). Thus N(x_{0},y_{0}) = (f_{x}(x_{0},y_{0}),f_{y}(x_{0},y_{0}),1)/√(f_{x}^{2}(x_{0},y_{0}) + f_{y}^{2}(x_{0},y_{0}) + 1).
Demos
Tangent Plane
 
For this demonstration, choose a point (x_{0},y_{0}) in the domain using the white hotspot in the window labeled "Domain: f(x,y)". The second window graphs the function f(x,y) and the x and yslice curves at the point f(x_{0},y_{0}). The tangent lines to each of the slice curves at f(x_{0},y_{0}) are also drawn, and it is clear that together they determine a plane. This plane, which is shown in yellow, is tangent to the graph at f(x_{0},y_{0}).

Intersection Set of Tangent Plane
 
This demo shows the tangent planes for the monkey saddle Ax^{3}  3Bxy^{2}. Note that the intersection set of the tangent plane with the function graph allows us to classify the points of the surface into two categories: for some points in the domain, the tangent plane meets a neighborhood of the point (x_{0},y_{0},f(x_{0},y_{0}) on the graph in a single point, whereas for other points, the tangent plane meets the graph in a pair of curves, or occasionally in a more complicated set of points.

Normal Vectors
 
This demo shows the unit normal vector to the tangent plane for a given point (x_{0},y_{0}) in the domain. Note that this vector is perpendicular to each of the normal lines drawn to the slice curves.

Exercises 1. Try entering f(x,y) = abs(abs(x)  abs(y))  abs(x)  abs(y). Using the limit definition of the partial derivatives, show that f_{x}(0,0) = f_{y}(0,0) = 0. Are f_{x}(0,0) and f_{y}(0,0) welldefined? Using the definintion of the tangent plane, show that the tangent plane is z(x,y) = 0. Move the hot spot to the origin. Is the plane z = 0 tangent to the surface? Note that to be tangent to a surface means to touch it at only one point.
2. Which is a stronger condition: the tangent plane exists at a point (x_{0},y_{0}); or, the partial derivatives are welldefined at a point (x_{0},y_{0})?
3. Use the second demo to classify the points of the graphs of the function f(x,y) = Ax^{2} + 2Bxy + Cy^{2} for different values of A, B and C.
4. Now change the function f(x,y) in the demo and classify the points of the functions f(x,y) = x^{2} + y^{2}, g(x,y) = x^{2}  y^{2}, and h(x,y) = x^{4} + 2x^{2}  y^{2}.
