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Gauss's Theorem / The Divergence Theorem 3. Consider the vector field F(x, y, 2) = (x², xy, y) and the region M bounded by the planes z = 0, x = 0, y = 0, and z = 4 - 2x – y. M might best be described as an irregular pyramid, bounded by four triangles that will be collectively referred to as T. So T is the 20 surface of the 3D volume M. Unfortunately, one of your tasks in this problem is to find the flux through T, which means integrating over each of those four triangles separately. Sorryl Hint: you'll soon discover that two of those four fluxes are zero. a) Graph the plane z = 4- 2x - y to help us get our bearings. You are welcome to graph the other three planes also, but they are just the three coordinate planes, so it's easy to see them without explicitly graphing them. Having all four planes graphed at once looks more confusing, in my opinion. b) Find V. F, the divergence of F. You will use this below. Now our goal is to verify the Divergence Theorem, though only once this time. The Divergence Theorem claims that V. FaV = |F. ds c) First evaluate the leftmost expression directly, the volume integral of the divergence of Fover the interior of the pyramid M. This part has some unpleasant algebra but otherwise is relatively simple.

Gauss's Theorem / The Divergence Theorem 3. Consider the vector field F(x, y, 2) = (x², xy, y) and the region M bounded by the planes z = 0, x = 0, y = 0, and z = 4 - 2x – y. M might best be described as an irregular pyramid, bounded by four triangles that will be collectively referred to as T. So T is the 20 surface of the 3D volume M. Unfortunately, one of your tasks in this problem is to find the flux through T, which means integrating over each of those four triangles separately. Sorryl Hint: you'll soon discover that two of those four fluxes are zero. a) Graph the plane z = 4- 2x - y to help us get our bearings. You are welcome to graph the other three planes also, but they are just the three coordinate planes, so it's easy to see them without explicitly graphing them. Having all four planes graphed at once looks more confusing, in my opinion. b) Find V. F, the divergence of F. You will use this below. Now our goal is to verify the Divergence Theorem, though only once this time. The Divergence Theorem claims that V. FaV = |F. ds c) First evaluate the leftmost expression directly, the volume integral of the divergence of Fover the interior of the pyramid M. This part has some unpleasant algebra but otherwise is relatively simple.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 60E
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Gauss's Theorem / The Divergence Theorem 3. Consider the vector field F(x,y, 2) = (x², xy, y) and the region M bounded by the planes z = 0, x = 0, y = 0, and z = 4 – 2x – y. M might best be described as an irregular pyramid, bounded by four triangles that will be collectively referred to as T. So T is the 20 surface of the 3D volume M. Unfortunately, one of your tasks in this problem is to find the flux through T, which means integrating over each of those four triangles separately. Sorry! Hint: you'll soon discover that two of those four fluxes are zero. a) Graph the plane z = 4– 2x - y to help us get our bearings. You are welcome to graph the other three planes also, but they are just the three coordinate planes, so it's easy to see them without explicitly graphing them. Having all four planes graphed at once looks more confusing, in my opinion. b) Find V. F, the divergence of F. You will use this below. Now our goal is to verify the Divergence Theorem, though only once this time. The Divergence Theorem claims that V. FaV = || F- ds c) First evaluate the leftmost expression directly, the volume integral of the divergence of Fover the interior of the pyramid M. This part has some unpleasant algebra but otherwise is relatively simple. d) Now we must evaluate the right-hand side. This is a real pain to do, because as stated before, there are four distinct boundary triangles. The flux through each of those four triangles must be evaluated separately. That means you need to find the intersection lines of the diagonal plane with each of the three coordinate planes. It also means you need to parameterize each boundary plane and find its normal vector. Be very careful that you adjust the orientation of the normal vector for each of the four planes to be outward. You've done all of this before, but I wanted to be clear about what you're getting into! Your two values from parts c and d should match once again, verifying that the Divergence Theorem is true in this case!
Transcribed Image Text:Gauss's Theorem / The Divergence Theorem 3. Consider the vector field F(x,y, 2) = (x², xy, y) and the region M bounded by the planes z = 0, x = 0, y = 0, and z = 4 – 2x – y. M might best be described as an irregular pyramid, bounded by four triangles that will be collectively referred to as T. So T is the 20 surface of the 3D volume M. Unfortunately, one of your tasks in this problem is to find the flux through T, which means integrating over each of those four triangles separately. Sorry! Hint: you'll soon discover that two of those four fluxes are zero. a) Graph the plane z = 4– 2x - y to help us get our bearings. You are welcome to graph the other three planes also, but they are just the three coordinate planes, so it's easy to see them without explicitly graphing them. Having all four planes graphed at once looks more confusing, in my opinion. b) Find V. F, the divergence of F. You will use this below. Now our goal is to verify the Divergence Theorem, though only once this time. The Divergence Theorem claims that V. FaV = || F- ds c) First evaluate the leftmost expression directly, the volume integral of the divergence of Fover the interior of the pyramid M. This part has some unpleasant algebra but otherwise is relatively simple. d) Now we must evaluate the right-hand side. This is a real pain to do, because as stated before, there are four distinct boundary triangles. The flux through each of those four triangles must be evaluated separately. That means you need to find the intersection lines of the diagonal plane with each of the three coordinate planes. It also means you need to parameterize each boundary plane and find its normal vector. Be very careful that you adjust the orientation of the normal vector for each of the four planes to be outward. You've done all of this before, but I wanted to be clear about what you're getting into! Your two values from parts c and d should match once again, verifying that the Divergence Theorem is true in this case!
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