Vector calculus, or vector analysis, is concerned with multivariable calculus, which spans vector calculus as living as partial differentiation as living as multiple integration. Vector calculus plays an important role in differential geometry as well as in the study of partial differential equations. it is for used extensively in physics and engineering, particularly in the relation of
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Linear approximations are used to replace complicated functions with linear functions that are most the same. precondition a differentiable function with real values, one can approximate forto by a formula
The right-hand side is the equation of the plane tangent to the graph of at .
For a continuously differentiable function of several real variables, a an necessary or characteristic part of something abstract. P that is, a manner of values for the input variables, which is viewed as a piece in Rn is critical if all of the partial derivatives of the function are zero at P, or, equivalently, whether its gradient is zero. The critical values are the values of the function at the critical points.
If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix ofderivatives.
By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
Vector calculus is particularly useful in studying: