Tangent planes
Section: Understanding Tangent Planes

A tangent plane is a plane that just ‘touches’ a surface at a particular point, without cutting into the surface. It provides an approximation of the surface near that point.

The tangent plane to a surface at a point is identical to the tangent plane to the level curve of the function at that point.

The concept of a tangent plane extends the idea of a tangent line to a function of two variables, instead of one.

Crucially, the direction of quickest change of the function at a point is represented by the vector normal (perpendicular) to the tangent plane.
Section: Constructing Tangent Planes

The equation of the tangent plane to a surface at a point can be found using partial derivatives.

If a function f(x, y) represents a surface, and P(a, b, f(a, b)) is a point on this surface, the equation of the tangent plane at P can be given as: f(x, y) = f(a, b) + f_x(a, b)(x  a) + f_y(a, b)(y  b).

The expressions f_x(a, b) and f_y(a, b) are the partial derivatives of the function f with respect to x and y respectively, evaluated at the point (a, b).
Section: Interpreting and Using Tangent Planes

Tangent planes allow us to make linear approximations to functions of several variables. This is useful when dealing with complex functions where exact values are difficult to calculate.

The accuracy of the approximation depends on how well the function can be approximated by a plane in the vicinity of the particular point.

Tangent planes can also be used to understand and illustrate the behaviour of functions, and solve problems in diverse scientific and mathematical fields.

As tangent planes are used to calculate gradients, you’ll often encounter tangent planes when dealing with optimisation problems.
Section: Application of Tangent Planes

The concept of tangent planes is critical in a variety of realworld applications, including physics, engineering, computer graphics, and more.

For example, plane tangents are used in the physics to study rates of change and in engineering for approximating the measurements.

In calculus, tangent planes play a vital role in problemsolving tools such as Newton’s method, used to find the zeros of a function (places where the function equals zero).