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Verifying Stokes’ Theorem Verify that the line
7. F = 〈x, y, z〉 ; S is the paraboloid z = 8 – x2 – y2, for 0 ≤ z ≤ 8, and C is the circle x2 + y2 = 8 in the xy-plane.
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Calculus: Early Transcendentals (2nd Edition)
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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?arrow_forwardConsider the surface given by the parametric vector function (image) 1. The graph of r (u, v) is the same graph of the surface z2 = x2 − y2 2. The surface is smooth in all its points. which is correct, incorrect or botharrow_forwardLet (P) be a plane considered as a surface in the space, parameterized by X(u, v) = (u, v, au + bv + c) where a, b, and c are all constants, with c + 0. Then: The tangent plane at each point is perpendicular to (P) The normal vector varies constantly The above answer The above a nswer The second fundamental form equals e The second fundamental form is zero The above answer The above a ns werarrow_forward
- Use either Stokes’ theorem or the divergence theorem to evaluate each of the followingintegrals in the easiest possible way.arrow_forward3. Consider the parametric vector equation of a cone r(u,v) = u cos vi + u sin vj + uk with 1arrow_forwardEvaluate 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_forward2. A cartesian equation for the surface is? 3. Draw the graph and the tangent planearrow_forwardUse Stokes' Theorem to evaluate ∫ C F · dr where F = (x + 5z) i + (7x + y) j + (2y − z) k and C is the curve of intersection of the plane x + 3y + z = 12 with the coordinate planes.(Assume that C is oriented counterclockwise as viewed from above.)arrow_forwardUse Stokes' Theorem to evaluate curl F- dS where F(r, y, 2) = 4ryzi – ryj + x?yzk and S consists of the top and the four sides (but not the bottom) of the cube with vertices (+2, ±2, +2), oriented outward. Since the box is oriented outwards the boundary curve must be transversed ? v when viewed from the top. A parametrization for the boundary curve C seen below from above can be given by: rT(t) rL(t) FR(t) rB(t) rz(t) = ( Σ Σ Σ 0arrow_forwardFind both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos(v)j+usin(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u , , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+ y+y+ z+z+ =0arrow_forwardExercise. The position vector for the curve C in the (, y, z)-space is given by r (t) = (2t + 1, t² , 3t). Find values for the constants a and b so the curve C lies on the surface x2 – 4y + az = b or explain why no such constants exist. Are there values for a and b for which the curve lies on the surface? yes ? Check work noarrow_forwardLet F = (y,3z + x,3y). Use Stokes' Theorem to find a plane with the equation ax + by + cz = 0 (where a, b, c are not all zero) such that f. F· dr = 0 for every closed C lying in the plane. Hint: Choose a, b, c so that the curl(F) lies in the plane. (Use symbolic notation and fractions where needed.) Choose the statement that describes a possible equation of the plane. Oc = 3a and b is arbitrary. O a, b, and c are arbitrary. Oc = 6a and b is arbitrary. O b = a and c is arbitrary. anacendentals Publisher: W.H. Freeman Question Source: Rogawski 4arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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