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.Surface Integral with respect to coordinate areaelements.Definition. Let g = g(x, y, z) be a scalar function defined on a domaincontaining S, define the surface integral of S with respect to thecoordinate x-axis, y-axis and z-axis respectively as∫∫∫∫∂(y, z)g(x, y, z)n · i dS = g(r(u, v)) du dv,SD ∂(u, v)∫∫∫∫∂(z, x)g(x, y, z)n · j dS = g(r(u, v)) du dv, andSD ∂(u, v)∫∫∫∫∂(x, y)g(x, y, z)n · k dS = g(r(u, v)) du dv,. SD ∂(u, v)n · i dS = 1 ( )∂(y, z)∥N∥ ∂(u, v) , · · · , · · · · (1, 0, 0) ∥N∥ du dv =∂(y, z)du dv = dx dy.∂(u, v)Remark. The jacobian factors are used to make the integralindependent of parametrization of the surface S.. . . . . .
.Surface Integral of Vector Field.Let F = (P, Q, R) be a continuous vector field defined in a domaincontaining a smooth oriented surface S, with unit normal vector field.Define the flux of F across S, or the surface integral of F over S as∫∫∫∫( )∂(y, z) ∂(z, x) ∂(x, y)F · n dS = (P, Q, R) · , , du dvS∫∫D∂(u, v) ∂(u, v) ∂(u, v)= P dydz + Q dzdx + R dxdy.. SRemark. If we change the orientation of S, the surface integral willchange by a minus sign.. . . . . .
Page 1 and 2: .Chapter 15 Vector Calculus.15.1 Ve
Page 3: .Definition. Let S be a parameteriz
Page 7: .Example. Let S be a parametric sur
Page 13 and 14: .Example. Find the flux of F(x, y,
Page 15 and 16: .Example. Suppose that a uniform so
Page 17 and 18: .Example. Let S be the surface (wit
Page 19 and 20: .Example. Let S be the sphere x 2 +
Page 21 and 22: .Example. Show that the divergence
Page 23 and 24: .Example. Let F(x, y, z) = (x, y, z
Page 25 and 26: .Stokes’ Theorem.Definition. A pi
Page 27 and 28: .Example. ∮ Let F(x, y, z) = 3zi
Page 29 and 30: .∫∫Example. Evaluate the surfac
Page 31 and 32: .Proposition. Let F be a continuous
Page 33 and 34: .Summary of Line Integral of Vector
Page 35 and 36: .Example. Let T be the solid bounde
Page 37: .Example. Let F(x, y, z) = f (x, y,

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