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Fundamental theorem of calculus for line integrals. Add languages. Add links. Article; ... Download as PDF; Printable version; ... Redirect page. Redirect to ...
Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use.
The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If φ : U ⊆ R n → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q , then
A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration. Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization ...
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy -plane bounded by the graph of f , the x -axis, and the lines x = a and x = b , such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total.
The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F ( x ) admits a derivative f ( x ) at all but countably many points, then f ( x ) is Henstock–Kurzweil integrable and F ( b ) − F ( a ) is equal to the integral of f on [ a , b ] .
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Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.