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...Surface integrals of vector fields lukas geyer montana state university m fall msu parametrized surfaces denition an orientation on a s is continuous choice unit normal en p at each point os example the xy plane has two orientations one given by e k pointing up n other down sphere kxk r outward x inward vectors obius strip caution not all are orientable most popular non depicted below integral component eld f oriented over zz ds it also called ux across or through applications flow rate uid with velocity magnetic and electric maxwell equations...