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In Einstein notation, the vector field has curl given by: where = ±1 or 0 is the Levi-Civita parity symbol. For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then ...
Differential calculus. The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point. In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1]
Elementary rules of differentiation. Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined [1][2] — including the case of complex numbers (C). [3]
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
Calculus. In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem.
e. In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation.
t. e. In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in ...
Calculus. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [1][2][3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. It is provable in many ways by using other derivative rules.
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