Editing Vector calculus (section)

=== Linear approximations ===
{{main|Linear approximation}}
Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function {{math|''f''(''x'', ''y'')}} with real values, one can approximate {{math|''f''(''x'', ''y'')}} for {{math|(''x'', ''y'')}} close to {{math|(''a'', ''b'')}} by the formula
: <math>f(x,y)\ \approx\ f(a,b)+\tfrac{\partial f}{\partial x} (a,b)\,(x-a)+\tfrac{\partial f}{\partial y}(a,b)\,(y-b).</math>

The right-hand side is the equation of the plane tangent to the graph of {{math|1=''z'' = ''f''(''x'', ''y'')}} at {{nowrap|{{math|(''a'', ''b'')}}.}}