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All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different. One generalization is to manifolds. In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This ...
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B ...
Variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full. The very broad use of variable changes is apparent when considering differential equations, where the independent variables may be changed using the chain rule or the dependent variables are changed resulting in ...
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V .
Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function ...
Although the Itô integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics. In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike the Itô calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds.
Big idea: use chain rule to compute rate of change of distance between two vehicles. Plan: Choose coordinate system; Identify variables; Draw picture; Big idea: use chain rule to compute rate of change of distance between two vehicles; Express c in terms of x and y via Pythagorean theorem; Express dc/dt using chain rule in terms of dx/dt and dy/dt