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The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

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Stoke's Theorem: The value of the line integral along a simple, closed, piecewise-smooth, positively oriented curve C, is related to the area of the surface C encloses. F must have continuous partial derivatives on a region in ℝ³. Stoke's theorem can be used to find the magnetic field strength a given distance from a straight wire (Ampere's law). C would represent the circumference of an imaginary circle at a constant distance around the wire, and the right side of the equation would be…

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The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its displacement vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral.

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Ballet and Dance Posters for Motion Theater by Caroline Grohs |

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Miele Kitchen. I spy a decanter (which always comes in handy). Clean lines. Marble counter top. Built in wine cabinet. Lovely.

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Magazine centrefold showing different versions of the Spitfire painted in the markings of various squadrons and air forces. Profiles of this type provide invaluable reference for researchers and aircraft modellers. Designed by Reginald Mitchell and powered by a Rolls-Royce Merlin engine, the Spitfire was integral to Britain's front-line air defence during World War II, and together with the Hawker Hurricane fighter played a key role in the Battle of Britain of 1940

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