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mathematics
calculus 6th edition

Calculus 6th Edition James Stewart - Solutions

Find the gradient vector field ∇f of f and sketch it.f(x, y) = x2 – y
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c x sin(y + z) ds, where C has parametric equations x = t2, y = t3, z = t4, 0 ≤ t ≤ 5
Find the gradient vector field ∇f of f and sketch it.f(x, y) = √x2 + y2
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c ze–xy ds, where C has parametric equations x = t, y = t2, z = e–t, 0 ≤ t ≤ 1
Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other.f(x, y) = sin x + sin y
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = e–xi + exj + ezk, C is the boundary of the part of the plane 2x + y + 2z = 2 in the first octantData from Stokes TheoremLet S be an oriented piecewise-smooth surface that is
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈u + v, u2, v2〉, –1 ≤ u ≤ 1, –1 ≤ v ≤ 1
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫c xe–2x dx + (x4 + 2x2y2) dy, C is the boundary of the region between the circles x2 + y2 = 1 and x2 + y2 = 4Data from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = x kFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = x3yi – x2y2 j – x2yz k, S is the surface of the solid bounded by the hyperboloid x2 + y2 – z2 = 1 and the planes z = –2 and z = 2Data from the Divergence
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (In y + 2xy3)i + (3x2y2 + x/y) j
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = yz i + 2xzj + exyk, C is the circle x2 + y2 = 16, z = 5Data from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple, closed,
Evaluate the line integral, where C is the given curve.∫c xyz ds. C: x = 2 sin t, y = t, z = –2 cos t, 0 ≤ t ≤ π
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = xy sin zi + cos(xz)j + y cos z k, S is the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1Data from the Divergence TheoremLet E be a simple solid region and let S be the boundary
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = j – iFigure 5Figure 9 y F (0, 3) 0 F (2,2) F (1,0) X
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = (xy cosh xy + sinh xy)i + (x2 cosh xy) j
If F = Pi + Qj, how do you test to determine whether F is conservative? What if F is a vector field on R3?
Use Stokes Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = xyi + 2zj + 3yk, C is the curve of intersection of the plane x + z = 5 and the cylinder x2 + y2 = 9Data from Stokes TheoremLet S be an oriented piecewise-smooth surface that is
Evaluate the line integral, where C is the given curve.∫c xyz2 ds. C is the line segment from (–1, 5, 0) to (1, 6, 4)
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.r(u, v) = 〈cos u sin v, sin u sin v, cos v + In tan(v/2)〉, 0 ≤ u ≤ 2π, 0.1 ≤ v ≤ 6.2
Evaluate the surface integral.S is the part of the cone z2 = x2 + y2 that lies between the planes z = 1 and z = 3 JJS x²z² ds.
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z)=x4i – x3z2j + 4xy2zk, S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = x + 2 and z = 0Data from the Divergence TheoremLet E be a
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈1, sin y〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = x2i + y2j. C is the are of the parabola y = 2x2 from (–1, 2) to (2, 8)
Evaluate the line integral, where C is the given curve.∫c (2x + 9z) ds, C: x = t y = t2, z = t3, 0 ≤ t ≤ 1
Evaluate the surface integral.∫∫s zds S is the surface x = y + 2z2, 0 ≤ y ≤ 1,0 ≤ z ≤ 1
Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant.x = u sin u cos v, y = u cos u cos v, z = u sin v
Use the Divergence Theorem to calculate the surface integral ∫∫S F · ds; that is, calculate the flux of F across S.F(x, y, z) = 4x3zi + 4y3zj + 3z4k, S is the sphere with radius R and center the originData from the Divergence TheoremLet E be a simple solid region and let S be the boundary
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (ex + x2y, ey – xy2), C is the circle x2 + y2 = 25 oriented clockwiseData from Green's TheoremLet C be a positively oriented, piecewise-smooth, simple closed curve in the
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈x – 2, x + 1〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = xy2i + x2yj, C: r(t) = 〈t + sin1/2πt, t + cos1/2πt〉, 0 ≤ t ≤ 1
Verify that Stokes Theorem is true for the given vector field F and surface S.F(x, y, z) = y2i + xj + z2k, S is the part of the paraboloid z = x2 + y2 that lies below the plane z = 1, oriented upwardData from Stokes TheoremLet S be an oriented piecewise-smooth surface that is bounded by a simple,
Evaluate the line integral, where C is the given curve.∫c x2y√z dz, C: x = t3, y = t, z = t2, 0 ≤ t ≤ 1
Use Green’s Theorem to evaluate ʃC F · dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = 〈y – In(x2 + y2), 2 tan–1(y/x)〉, C is the circle (x – 2)2 + (y – 3)2 = 1 oriented counterclockwiseData from Green's TheoremLet C be a positively oriented,
Match the vector fields F with the plots labeled I–IV. Give reasons for your choices.F(x, y) = 〈y, 1/x〉 I ترا III -3 1 1 " . 1 1 F 3 -3 3 NIZ. -3 3 برا II ch IV ņ 5 S 5 1/12 5 5
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y) = y2/1 + x2 i + 2y arctan x j, C: r(t) = t2i + 2t j, 0 ≤ t ≤ 1
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = xyz2 i + x2yz2j + x2y2z k
Evaluate the line integral, where C is the given curve.∫c z dx + x dy + y dz, C: x = t2, y = t3, z = t2, 0 ≤ t ≤ 1
Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral.P(x, y) = y2ex, Q(x, y) = x2ey, C consists of the line segment from (–1, 1) to (1, 1) followed by the arc of the parabola y = 2 – x2 from (1, 1) to (–1, 1)Data from Green's
Evaluate the surface integral.∫∫S y2dS is the part of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane
Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices.F(x, y, z)= i + 2j + 3 k I 1 z0 -1 8 -0 z -1- -1 01 y -1 0 II 0 Z 1 -1 IV z 0 -1 -1 0 1 10-1 x
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constantr(u, v) = sin vi + cos u sin 2v j + sin u sin 2v k II N 8MM N III 3 N A
Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices.F(x, y, z)= i + 2j + zk I 1 z0 -1 8 -0 z -1- -1 01 y -1 0 II 0 Z 1 -1 IV z 0 -1 -1 0 1 10-1 x
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = 2xy i + (x2 + 2yz)j + y2k
Evaluate the line integral, where C is the given curve.∫c (x + yz) dx + 2x dy + xyz dz, C consists of line segments from (1, 0, 1) to (2, 3, 1) and from (2, 3, 1) to (2, 5, 2)
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = (2xz + y2)i + 2xyj + (x2 + 3z2) k, C: x = t2, y = t + 1, z = 2t – 1, 0 ≤ t ≤ 1
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z)= ez i + j + xez k
Evaluate the line integral, where C is the given curve.∫c x2 dx + y2 dy + z2 dz, C consists of line segments from (0, 0, 0) to (1, 2, –1) and from (1, 2, –1) to (3, 2, 0)
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = y2 cos z i + 2xy cos z j – xy2 sin z k, C: r(t) = t2i + sin tj + tk, 0 ≤ t ≤ π
Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constantx = (1 –|u|) cos v, y = (1 – |u|) sin v, z = u N II 8MM N III 3 N A
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = ye–xi + e–x j + 2zk
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F • dr along the given curve C.F(x, y, z) = eyi + xeyj + (z + 1)ezk, C: r(t) = ti + t2j + t3k, 0 ≤ t ≤ 1
Find a parametric representation for the surface.The plane that passes through the point (1, 2, –3) and contains the vectors i + j – k and i – j + k
Show that the line integral is independent of path and evaluate the integral.∫c tan y dx + x sec2y dy, C is any path from (1, 0) to (2, π/4)
Find a parametric representation for the surface.The lower half of the ellipsoid 2x2 + 4y2 + z2 = 1
Is there a vector field G on R3 such that curl G = 〈xyz, – y2z, y2z〉? Explain.
Show that, under conditions to be stated on the vector fields F and G, curl(F x G) = F div G - G div F + (GV)F- (F.V)G
Show that the line integral is independent of path and evaluate the integral.∫c (1 – ye–x) dx + e–xdy, C is any path from (0, 1) to (1, 2)
Plot the vector field and guess where div F > 0 and where div F < 0. Then calculate div F to check your guess.F(x, y) = 〈xy, x + y2〉
Find the gradient vector field of f.f(x, y) = xexy
Find a parametric representation for the surface.The part of the hyperboloid x2 + y2 – z2 = 1 that lies to the right of the xz-plane
Find the work done by the force field F in moving an object from P to Q. lF(x, y) = 2y3/2 i + 3x√y j; P(1, 1), Q(2, 4)
Find the gradient vector field of f.f(x, y) = tan(3x – 4y)
Find a parametric representation for the surface.The part of the elliptic paraboloid x + y2 + 2z2= 4 that lies in front of the plane x = 0
Find the work done by the force field F in moving an object from P to Q. F(x, y) = e–yi – xe–yj; P(0, 1), Q(2, 0)
Use a calculator or CAS to evaluate the line integral correct to four decimal places.∫c F · dr, where F(x, y) = xy i + sin y j and r(t) = eti + e–t2j, 1 ≤ t ≤ 2
Find the gradient vector field of f.f(x, y, z) = x cos(y/z)
Find a parametric representation for the surface.The part of the sphere x2 + y2 + z2 = 16 that lies between the planes z = –2 and z = 2
Find a parametric representation for the surface.The part of the cylinder y2 + z2 = 16 that lies between the planes x = 0 and x = 5
Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other.f(x, y) = sin(x + y)
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) |x > 0, y > 0}
Find parametric equations for the surface obtained by rotating the curve y = e–x, 0 ≤ x ≤ 3, about the x-axis and use them to graph the surface.
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) | x ≠ 0}
Find parametric equations for the surface obtained by rotating the curve x = 4y2 – y4, –2 ≤ y ≤ 2, about the y-axis and use them to graph the surface.
Evaluate ∫∫∫S xyz ds correct to four decimal places, where S is the surface z = xy, ≤ 0 ≤ 1, 0 ≤ y ≤ 1.
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) |1 < x2 + y2 < 4}
Find the exact value of ∫∫∫S x2yz ds, where S is the surface in Exercise 31.Data from Exercises 31Evaluate ∫∫∫S xyz ds correct to four decimal places, where S is the surface z = xy, ≤ 0 ≤ 1, 0 ≤ y ≤ 1.
Let(a) Show that ∂P/∂y = ∂Q/∂x.(b) Show that ∫c F · dr is not independent of path. [Compute ∫c1 F · dr and ∫c2 F · dr where C1 and C2 are the upper and lower halves of the circle x2 + y2 = 1 from (1, 0) to (–1, 0).] Does this contradict Theorem 6?Data from Theorem 6Let F = Pi +
Determine whether or not the given set is (a) Open,(b) Connected, (c) Simply-connected.{(x, y) | x2 + y2 ≤ 1 or 4 x2 + y2 ≤ 9}
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane.x = u + v, y = 3u2, z = u – v; (2, 3, 0)
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane.x = u2, y = v2, z = uv; u = 1, v = 1
Find an equation of the tangent plane to the given parametric surface at the specified point. If you have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane.r(u, v) = uvi + u sin vj + v cos uk; u = 0, v = π
Find the area of the surface.The part of the plane 2x + 5y + z = 10 that lies inside the cylinder x2 + y2 = 9
Find the work done by the force field F(x, y) = x sin yi + yj on a particle that moves along the parabola y = x2 from (–1, 1) to (2, 4).
Find the work done by the force field F(x, y, z) = 〈y + z,x + z,x + y〉 on a particle that moves along the line segment from (1, 0, 0) to (3, 4, 2).
Find the area of the surface.The part of the surface z = 1 + 3x + 2y2 that lies above the triangle with vertices (0, 0), (0, 1), and (2, 1)
Find the area of the surface.The part of the paraboloid x = y2 + z2 that lies inside the cylinder y2 + z2 = 9
Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.The part of the surface z = e–x2–y2 that lies above the disk x2 + y2 ≤ 4
Find the area of the surface.The part of the surface y = 4x + z2 that lies between the planes x = 0, x = 1, z = 0, and z = 1
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = x sin(x + 2y)
Verify that the conclusion of Clairaut's Theorem holds, that is, uxy = uyx.u = x4y2 – 2xy5
Sketch the graph of the function.f(x, y) = 1 – y2
Use the Chain Rule to find dz/dt or dw/dt.z = √1 + x2 + y2, x = ln t, y = cos t
Find an equation of the tangent plane to the given surface at the specified point.z = √xy, (1, 1, 1)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).kf(x, y) = x2 + y2; xy = 1
Sketch the graph of the function.f(x, y) = x2 + (y – 2)2
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f(x, y) = x2y3 – y4, (2, 1), θ = π/4
Use the Chain Rule to find dz/dt or dw/dt.z = tan–1(y/x), x = et, y = 1 – e–t
Find an equation of the tangent plane to the given surface at the specified point.z = y ln x, (1, 4, 0)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y) = 4x + 6y; x2 + y2 = 13

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