tangentplan

A tangent plane is a fundamental concept in differential geometry and multivariable calculus that approximates a surface near a given point. It serves as the linear analog of a tangent line to a function of a single variable. For a twice-differentiable surface defined by the equation \( F(x, y, z) = 0 \), the tangent plane at a point \( (x_0, y_0, z_0) \) on the surface can be derived using the gradient of \( F \). The equation of the tangent plane is given by:

\[ F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0, \]

where \( F_x, F_y, \) and \( F_z \) represent the partial derivatives of \( F \) with respect to \( x, y, \) and \( z \), respectively.

For a function \( z = f(x, y) \), the tangent plane at \( (x_0, y_0, f(x_0, y_0)) \) can be expressed as:

\[ z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0), \]

where \( f_x \) and \( f_y \) are the partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively. The tangent plane provides a linear approximation of the surface near the point and is used in various applications, including optimization, physics, and engineering.

Geometrically, the tangent plane touches the surface only at the given point but closely approximates the surface in its immediate vicinity. The concept extends to higher dimensions, where a hyperplane serves as the analogous linear approximation for a hypersurface. The tangent plane is also crucial in defining the differential of a function, which is a linear map that approximates small changes in the function’s output. This approximation becomes exact in the limit as the changes approach zero, aligning with the definition of the derivative in multivariable calculus.