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Note 5. Surface Integrals • Parametric equations of surfaces A surface in space can be represented by parametric equations x = x(u, v), y = y(u, v), z = z(u, v), (u, v) ∈ Q, or equivalently, by the vector equation r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k, (u, v) ∈ Q, where Q is a region in the uv-plane. A unit normal vector of the surface is given by ∂y ∂y ∂x ∂z ∂z n = ru × rv /|ru × rv |, where ru = ∂u i + ∂u j + ∂u k and rv = ∂x ∂v i + ∂v j + ∂v k. • Surface integrals with respect to surface area If f is a continuous function on S, then ZZ ZZ f (x, y, z) dS = f x(u, v), y(u, v), z(u, v) ru × rv dA. S Q In particular, when f = 1, the above surface integral gives the area of the surface. • Surface integrals of vector fields If F is a continuous vector field on an oriented surface S with unit normal vector n, then the surface integral of F over S, or the flux of F across S, is ZZ ZZ F·n dS = F·(ru × rv ) dA. S Q • The Divergence Theorem The divergence of a vector field F = P i + Q j + R k is defined by ∂P ∂Q ∂R + + , div F = ∇ · F = ∂x ∂y ∂z ∂ ∂ ∂ where ∇ denotes the operator ∂x i + ∂y j + ∂z k. Let E be a solid region whose boundary surface S has outward orientation. Let F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. Then ZZ ZZZ F·n dS = div F dV. S E • The Stokes Theorem The curl of a vector field F = P i + Q j + R k is defined by ∂P ∂R ∂Q ∂P ∂R ∂Q − i+ − j+ − k. curl F = ∇ × F = ∂y ∂z ∂z ∂x ∂x ∂y Let S be an oriented piecewise smooth surface that is bounded by a simple, closed, piecewise smooth boundary curve C with the orientation induced by the orientation of S. Then Z ZZ F·dr = (curl F)·n dS. C S