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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."
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
Example 1a. The function is f(x, y) = (x − 1) 2 + √ y; if one adopts the substitution u = x − 1, v = y therefore x = u + 1, y = v one obtains the new function f 2 (u, v) = (u) 2 + √ v. Similarly for the domain because it is delimited by the original variables that were transformed before (x and y in example)
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)⋅C, nor from A⋅(B ...
Composable differentiable functions f : R n → R m and g : R m → R k satisfy the chain rule, namely () = (()) for x in R n. The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix , which in a sense is the " second derivative " of the function in question.
When u and v are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values. Pythagorean theorem : It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides.
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where a ( x ) = a ∈ R {\displaystyle a(x)=a\in \mathbb {R} } is constant, b ( x ) = x , {\displaystyle b(x)=x,} and f ( x , t ...
Using unit vectors for u and/or v, individual columns, rows or elements [4] of A may be manipulated and a correspondingly updated determinant computed relatively cheaply in this way. When the matrix determinant lemma is used in conjunction with the Sherman–Morrison formula, both the inverse and determinant may be conveniently updated together.