Editing Tangent (section)

==Tangent plane to a surface{{anchor|For surfaces|Surfaces|Plane}}==
{{redirect|Tangent plane|the geographical concept|Local tangent plane}}
{{further|Differential geometry of surfaces#Tangent plane|Parametric surface#Tangent plane}}
{{see also|Normal plane (geometry)}}The '''tangent plane''' to a [[Surface (geometry)|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''. Mathematically, if the surface is given by a function <math>z = f(x, y)</math>, the equation of the tangent plane at point <math>(x_0, y_0, z_0)</math> can be expressed as:

<math>z-z_0 = \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0)</math>.

Here, <math display="inline">\frac{\partial f}{\partial x}</math> and <math display="inline">\frac{\partial f}{\partial y}</math> are the partial derivatives of the function <math>f</math> with respect to <math>x</math> and <math>y</math> respectively, evaluated at the point <math>(x_0, y_0)</math>. In essence, the tangent plane captures the local behavior of the surface at the specific point ''p''. It's a fundamental concept used in calculus and differential geometry, crucial for understanding how functions change locally on surfaces.