algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of: Also commonly used are the two triple products: Operators and theorems Differential operators Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (), also known as "nabla". The three basic vector operators are: Also commonly used are the two Laplace operators: A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration. Integral theorems The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions: In two dimensions, the divergence and curl theorems reduce to the Green's theorem: Applications Linear approximations Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function with real values, one can approximate for close to by the formula The right-hand side is the equation of the plane tangent to the graph of at Optimization For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rn) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if 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 of second derivatives. 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. Physics and engineering Vector calculus is particularly useful in studying: Center of mass Field theory Kinematics Maxwell's equations Generalizations Different 3-manifolds Vector calculus is initially defined for Euclidean 3-space, which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion