By David Dugdale (auth.)

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A scalar quantity can be specified by a single number that carries no reference to any direction in space. A vector quantity, by contrast, is only fully described if both its magnitude and associated direction in space are given. Both specifications are usually made by giving the components of the vector with respect to some coordinate system. Numerical values of these components are of course very much dependent on the choice of the nature and orientation of the coordinate system, in a way that the numerical value of a scalar is not.

6 (a) A plane surface, S" having a rectangular boundary. (b) A hemispherical surface, s2. sum the contributions from all such strips. The mathematical statement of this procedure is = f:==0L 2 (~~ +y 2 L 1 )dy=~(L~L 2 +L~L 1 ) The integration could equally well be done by integrating firstly over y at constant x to get the contributions from strips of width dx, followed by integration over x to sum all these strips. Any other way of covering this area would do equally well. Provided that the integration sums every element of area once and only once, then it is immaterial to the final result in what order this sum is performed.

Components of field tangential to the surface do not contribute to the surface integral. From this it follows that surface integrals of fields like the velocity flow v of fluids or the electric current density J in electromagnetism have a very direct physical significance. An integral over some surface is, in the fluid case, a measure of the flow of matter through that surface. In the electric case it measures the flow of electric charge through that surface. 3 Volume integrals The integration of fields through volumes can also give results having direct physical significance.