# D r a curve joining p 0 to ab f d r 1 h the line

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d r - a curve joining p 0 to ( a,b ) F · d r ) = 1 h ( the line joining ( a,b ) to ( a + h,b ) F · d r ) = 1 h h 0 ( f ( a + t, b ) , g ( a + t, b )) · (1 , 0) dt = 1 h h 0 f ( a + t, b ) dt which tends to d ds s 0 f ( a + t, b ) dt | s =0 as h tends to 0. This derivative is just f ( a, b ) by the fundamental theorem of calculus. Thus, f = ∂ϕ ∂x . Similarly, g = ∂ϕ ∂y and we conclude that F = ϕ .
4.4 Surface Integrals Definition 4.4.1. Let f be a function in 3 variables. S is a surface parametrized by r ( u, v ) for ( u, v ) in a certain region R . We define the surface integral of f over S to be S fdσ = R f ( r ( u, v )) r ∂u × r ∂v dudv. Remark 4.4.1. If a piece of metal plate occupies the surface S and f ( x, y, z ) =the density of the metal plate at the point ( x, y, z ). Then S fdσ =total mass of the metal plate. Remark 4.4.2. The surface integral of a function over a surface is independent of the parametrization chosen for the surface. However, we are unable to prove it here for the lack of a change of variable formula. Example 4.4.1. Let f ( x, y, z ) = x 2 + y 2 and let S be the unit northern hemisphere. Evaluate S fdσ .
42 Remark 4.4.4. We saw that n = r ∂u × r ∂v r ∂u × r ∂v is indeed an unit normal to S , while r ∂u × r ∂v dudv is the surface area element of S . Therefore, the flux integral of F over S is indeed the surface integral of F · n over S . Thats why such a symbol is adopted. However, there are always two choices for an unit normal to the surface. We make the following convention when dealing with closed surfaces (i.e. a surface that divides the whole space into two parts: the interior and exterior.), we always make the choice of the unit outward normal. Remark 4.4.5. Consider the case when we are studying the sea. F ( x, y, z ) is the flow velocity at the point ( x, y, z ). Let S be an imaginary surface in the sea. Then, S F · n is the volume of water flowing THROUGH S per unit time. In case S is a closed surface and R is the interior of S , - S F · n is the volume of water accumulated in R per unit time.