# Calculus 2a Real Functions in Several Variables Guidelines by Leif Mejlbro

By Leif Mejlbro

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Com 43 Calculus 2a Line integrals 15. Line integrals This is a 1-dimensional case where the domain of integration is not an interval, but a curve in the plane or the space. The reduction formula becomes b f (x) ds = K f (r(t)) r (t) dt. 5 1 1 Figure 11: The curve of parameter representation r(t) = (cos t, sin t, t). Procedure: 1) Write down a rectangular parameter representation for the curve K: In the plane: (x, y) = r(t), In the space: (x, y, z) = r(t), t ∈ [a, b]. t ∈ [a, b]. 2) Calculate the curve element (the weight is r (t) ) ds = r (t) dt.

General: x = X(u, v, w), y = Y (u, v, w), z = Z(u, v, w), injective almost everywhere. These are our building stones in the ongoing geometrical analysis. 1 If the exercise does not contain any hint, then choose among the rectangular, polar, semi-polar or spherical coordinates. On the other hand, if more general coordinates are needed in an exercise, then these will always be given, usually in their inverse form: dimension 2 : u = U (x, y), v = V (x, y), dimension 3 : u = U (x, y, z), v = V (x, y, z), w = W (x, y, z).

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