3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V₁, V2, V3) : a ≤ v₁ ≤ b, f(v₁) ≤ v2 ≤ g(v₁), 6(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b = R are constants, f, g are functions of v₁ and 6, are functions of v₁, v2. (a) The region outside the cylinder x² + y² ≤ ² and inside the sphere x² + y² + z² = R² assuming r< R. Express this in the cylindrical and spherical coordinate systems. (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.
3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V₁, V2, V3) : a ≤ v₁ ≤ b, f(v₁) ≤ v2 ≤ g(v₁), 6(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b = R are constants, f, g are functions of v₁ and 6, are functions of v₁, v2. (a) The region outside the cylinder x² + y² ≤ ² and inside the sphere x² + y² + z² = R² assuming r< R. Express this in the cylindrical and spherical coordinate systems. (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Express the following regions in the stated coordinate systems. You do not need to justify your answer
but showing your though process may help with partial marks.
Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤
v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any
order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2.
(a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z²
r< R. Express this in the cylindrical and spherical coordinate systems.
-
R² assuming
(b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the
cylindrical, spherical and Cartesian coordinate system.
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