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Verify Formula (1) in the Divergence Theorem by evaluating the surface
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EBK CALCULUS EARLY TRANSCENDENTALS SING
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Calculus: Early Transcendentals (3rd Edition)
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- Evaluate F.ndS for the given F and ơ. (b) F(x, y, z) = (x² + y) i+ xyj – (2xz + y) k, o : the surface of the plane x + y + z = 1 in the first octantarrow_forwardExplain briefly how Vø relates to surfaces of constant (x, y, x). Hence, use Vø to find points in (x, y, z) where the surface xy + yz + zx + 2y + 2z = -c² is parallel to the xy plane.arrow_forwardExplain why C lies on the surface of the cylinder x² + y² = 1.arrow_forward
- Consider the surface that can be parameterized as for u, v = [0, 2π). x(u, v) y (u, v) z (u, v) = = = COS V (u² - 1) (u² − 1) sin v U Let x¹ u and ² v. Find the line clement for the surface. What is the metric tensor and the dual metric tensor? (b) (c) (d) What is the value of the component R212 of the Riemann curvature tensor? Make sure you simplify your answer. Determine the values of all the Christoffel coefficients of the surface.arrow_forward6. Use the Stokes theorem to evaluate f Fdr where F (3x – 2y +32) i +(4x – 2y+32) 5 + (2x – 3y | 22) k and C is the circle: æ = 3, y? | 22 = 4, oricnted counterclockwisc when vicwcd from the positive part of x-axis. Describe the surface S whose boundary is C and state its orientation.arrow_forwardDescribe and sketch the surfaces in space (R³) defined by the following equa- tions: (a) y = -z +1 (b) x² + y? = 3arrow_forward
- Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (−1, 1, 1), (2, 1, −5), (2, 3, −11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1,-5) from (2, 1, 5) to (2, 3, -11) from (2, 3, -11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁ (t) = S₂(t) = S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) te [3, 4) UE [-1, 2], VE [1, 3]arrow_forwardLet S be the portion of the plane 2x + 3y + z = 3 lying between the points (-1, 1, 2), (2, 1, -4), (2, 3, -10), and (-1, 3, -4). Find parameterizations for both the surface S and its boundary as. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 2) to (2, 1, -4) S,(t) tE [0, 1) from (2, 1, -4) to (2, 3, -10) S2(t) tE[1, 2) from (2, 3, -10) to (-1, 3, -4) S,(t) = tE [2, 3) from (-1, 3, -4) to (-1, 1, 2) s,(t) tE [3, 4) boundary Ф(и, v) uE [-1, 2], v E [1, 3] IIarrow_forwardplease help with this!arrow_forward
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