Tangent Planes and Linear Approximations

The tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized.

Tangent Plane to a Sphere

The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.

Equations

When the curve is given by $y = f(x)$ the slope of the tangent is $\frac{dy}{dx}$, so by the point–slope formula the equation of the tangent line at $(x_0, y_0)$ is: 

$\frac{dy}{dx}(x_0,y_0) \cdot (x-x_0) - (y-y_0)$

where $(x, y)$ are the coordinates of any point on the tangent line, and where the derivative is evaluated at $x=x_0$.

The tangent plane to a surface at a given point $p$ is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at $p$, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to $p$ as these points converge to $p$. For a surface given by a differentiable multivariable function $z=f(x,y)$, the equation of the tangent plane at $(x_0,y_0,z_0)$ is given as:

$f_x(x_0,y_0) (x-x_0) + f_y(x_0,y_0) (y-y_0) - (z-z_0) = 0$

where $(x_0,y_0,z_0)$ is a point on the surface. Note the similarity of the equations for tangent line and tangent plane.

Linear Approximation

Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point. The approximation works well as long as the point $(x,y,z) $ under consideration is close enough to $(x_0,y_0,z_0)$, where the tangent plane touches the surface. The plane describing the linear approximation for a surface described by $z=f(x,y)$ is given as:

$z = z_0 + f_x(x_0,y_0) (x-x_0) + f_y(x_0,y_0) (y-y_0)$.