UNIVERSITY OF INDONESIA FACULTY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING

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1 UNIVERSITY OF INDONESIA FACULTY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING ENGINEERING MATHEMATICS (MCS-21007) 1. Course Name/Units : Engineering Mathematics/4 2. Department/Semester : Mechanical Engineering /3 3. Program/Period : S1/ First semester Date/Time/Room : I. Tuesday : (K105) : II. Thursday : (K206) 5. Instructor : 6. Topics : MCS : Divergence Theorem Divergence Theorem The Divergence Theorem When we looked at Greens Theorem, we saw that there was a relationship between a region and the curve that encloses it. This gave us the relationship between the line integral and the double integral. Moving to three dimensions, the divergence theorem provides us with a relationship between a triple integral over a solid and the surface integral over the surface that encloses the solid. The Divergence Theorem Let Q be a solid region bounded by a closed surface oriented with outward pointing unit normal vector N, and let F be a differentiable vector field (components have continuous partial derivatives). Then Example Find Where F(x,y,z) = y 2 i + e x (1-cos(x 2 + z 2 ))j + (x + z)k and S is the unit sphere centered at the point (1,4,6) with outwardly pointing normal vector. Solution This seemingly difficult problem turns out to be quite easy once we have the divergence theorem. We have divf = = 1 Now recall that a triple integral of the function 1 is the volume of the solid. Since the solid is a sphere of radius 1 we get 4/3. MCS Divergence Theorem- 1

2 Part of the Proof of the Divergence Theorem As usual, we will make some simplifying remarks and then prove part of the divergence theorem. We assume that the solid is bounded below by z = g1(x,y) and above by z = g1(x,y) Notice that the outward pointing normal vector is upward on the top surface and downward for the bottom region. We also note that the divergence theorem can be written as We will show that We have on the top surface Pk. N ds = Pk. ( (-g2)x i - (g2)y j + k) = P(x,y,g2(x,y)) On the bottom surface, we get Pk. N ds = Pk. ( (g1)x i + (g1)y j - k) = -P(x,y,g1(x,y)) Putting these together we get For the triple integral, the fundamental theorem of calculus tell us that MCS Divergence Theorem- 2

3 An Interpretation Of Divergence We have seen that the flux is the amount fluid flow per unit time through a surface. If the surface is closed, then the total flux will equal the flow out of the solid minus the flow in. Often in the solid there is a source (such as a star when the flow is electromagnetic radiation) or a sink (such as the earth collecting solar radiation) If we have a small solid S(P) containing a point P, then the divergence of the vector field is approximately constant, which leads to the approximation The divergence theorem expresses the approximation Flux through divf(p) (Volume) Dividing by the volume, we get that the divergence of F at P is the Flux per unit volume. If the divergence is positive, then the P is a source. If the divergence is negative, then P is a sink. Stokes' Theorem The divergence theorem is used to find a surface integral over a closed surface and Green's theorem is use to find a line integral that encloses a surface (region) in the xy-plane. The theorem of the day, Stokes' theorem relates the surface integral to a line integral. Since we will be working in three dimensions, we need to discus what it means for a curve to be oriented positively. Let S be a oriented surface with unit normal vector N and let C be the boundary of S. Then C is positively oriented if its orientation follows the right hand rule, that is if you right hand curls around N in the direction of C's orientation, then your thumb will be pointing in the direction of N. Now we are ready to state Stokes' Theorem. The proof will be left for a more advanced course. MCS Divergence Theorem- 3

4 Stokes' Theorem Let S be an oriented surface with unit normal vector N and C be the positively oriented boundary of S. If F is a vector field with continuous first order partial derivatives then Example Let S be the part of the plane z = 4 - x - 2y with upwardly pointing unit normal vector. Use Stokes' theorem to find Where F = yi + zj xyk Solution First notice that without Stokes' theorem, we would have to parameterize three different line segments. Instead we can find this with just one double integral. We have and N ds = i + 2j + k So that Curl F. N ds = 1 + x + 2y - 1 = x 2y We integrate Curl and Circulation Just as the divergence theorem assisted us in understanding the divergence of a function at a point, Stokes' theorem helps us understand what the Curl of a vector field is. Let P be a point on the surface and C e be a tiny circle around P on the surface. The MCS Divergence Theorem- 4

5 measures the amount of circulation around P. You can see this by noticing that if F flows in the direction of the tangent vector, then F. dr will be positive. If it flows in the opposite direction, then it will be negative. The stronger the force field in the direction of the tangent vector, the greater the circulation. Since the region enclosed by C e is tiny, the surface integral can be approximated by or Curl F. N = Circulation per unit area So the curl tell us how much the force field rotates around the point. We can see that if this is a small piece of the surface containing P, then Curl F. N > 0 MCS Divergence Theorem- 5