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mathematics
precalculus

Questions and Answers of Precalculus

If a is a constant vector, r = x i + y j + z k, and S is an oriented, smooth surface with a simple, closed, smooth, positively oriented boundary curve C, show that 2a 2a · dS = (a X r) · dr
If the components of F have continuous second partial derivatives and S is the boundary surface of a simple solid region, show that ∫∫S curl F • dS = 0.
Find ∫∫S F • n dS, where F(x, y, z) = x i + y j + z k and S is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed). ZA (0, 2, 2)
Let F(x, y) = (2x3 + 2xy2 - 2y) i + (2y3 + 2x3y + 2x) j/x2 + y2 Evaluate where C is shown in the figure. $ F• dr, х
Let F(x, y, z) = (3x2yz - 3y) i + (x3z - 3x) j + (x3y + 2z) k. Evaluate ∫C F • dr, where C is the curve with initial point (0, 0, 2) and terminal point (0, 3, 0) shown in the figure. z. ZA
Compute the outward flux of F(x, y, z) = x i + y j + z k/(x2 + y2 + z2 )3/2 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.
Use the Divergence Theorem to calculate the surface integral ∫∫s F • dS, where F(x, y, z) = x3 i + y3 j + z3 k and S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the
Use Stokes’ Theorem to evaluate ∫c F • dr, where F(x, y, z) = xy i + yz j + z x k, and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed
Use Stokes’ Theorem to evaluate ∫∫S curl F • dS, where F(x, y, z) = x2yz i + yz2 j + z3exy k, S is the part of the sphere x2 + y2 + z2 = 5 that lies above the plane z = 1, and S is oriented
Verify that Stokes’ Theorem is true for the vector field F(x, y, z) = x2 i + y2 j + z2 k, where S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-plane and S has upward
Evaluate the surface integral.∫∫S F • dS, where F(x, y, z) = x2 i + xy j + z k and S is the part of the paraboloid z = x2 + y2 below the plane z = 1 with upward orientation
Evaluate the surface integral. ∫∫S (x2z + y2z) dS, where S is the part of the plane z = 4 + x + y that lies inside the cylinder x2 + y2 = 4
Evaluate the surface integral.∫∫S F • dS, where F(x, y, z) = xz i - 2y j + 3x k and S is the sphere x2 + y2 + z2 = 4 with outward orientation
Evaluate the surface integral. ∫∫S z dS where S is the part of the paraboloid z = x2 + y2 that lies under the plane z = 4
(a) Find an equation of the tangent plane at the point (4, -2, 1)to the parametric surface S given by r(u, v) = v2 i - uv j + u2 k 0 < u < 3, -3 < v < 3(b) Use a computer to graph the
Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 2).
(a) Sketch the curve C with parametric equationsx = cos t y = sin t z = sin t 0 < t < 2π(b) Find
If f is a harmonic function, that is, ∇2f = 0, show that the line integral ∫fy dx - fx dy is independent of path in any simple region D.
If f and t are twice differentiable functions, show that∇2(fg) = f∇2g + g∇2f + 2∇f • ∇g v°(fg) = fV°g + gV²f + 2Vf • Vg
If C is any piecewise-smooth simple closed plane curve and f and g are differentiable functions, show that∫C f (x) dx + g(y) dy = 0.
If F and G are vector fields whose component functions have continuous first partial derivatives, show thatcurl(F x G) = F div G - G div F + (G • ∇)F - (F • ∇)G
Show that there is no vector field G such thatcurl G = 2x i + 3yz j - xz2 k
Find curl F and div F ifF(x, y, z) = e-x sin y i + e-y sin z j + e-z sin x k
Use Green’s Theorem to evaluate ∫Cx2y dx - xy2 dy, where C is the circle x2 + y2 = 4 with counterclockwise orientation.
Use Green’s Theorem to evaluate`where C is the triangle with vertices (0, 0), (1, 0), and (1, 3). VI + x³ dx + 2xydy
Verify that Green’s Theorem is true for the line integral ∫Cxy2 dx - x2 y dy, where C consists of the parabola y = x2 from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1).
Show that F is conservative and use this fact to evaluate ∫C F • dr along the given curve.F(x, y, z) = ey i + (xey + ez) j + yez k, C is the line segment from (0, 2, 0) to (4, 0, 3)
Show that F is conservative and use this fact to evaluate ∫C F • dr along the given curve.f(x, y ) = (4x3y2 - 2xy3)i + (2x4y - 3x2y2 + 4y3)j,C: r (t) = (t + sin πt)i + (2t + cos πt)j, 0 < t
Show that F is a conservative vector field. Then find a function f such that F = ∇f.F(x, y, z) = sin y i + x cos y j - sin z k
Show that F is a conservative vector field. Then find a function f such that F = ∇f .F(x, y) = (1 + xy)exy i + (ey + x2exy) j
Find the work done by the force field F(x, y, z) = z i + x j + y k in moving a particle from the point (3, 0, 0) to the point (0, π/2, 3) along (a) a straight line(b) the helix x = 3 cos t, y =
Evaluate the line integral.∫C F • dr, where F(x, y, z) = ez i + xz j + (x + y) k and C is given by r(t) = t2 i + t3 j - t k, 0 < t < 1
Evaluate the line integral.∫C F • dr, where F(x, y) = xy i + x2 j and C is given byr(t) = sin t i + (1 + t) j, 0 < t < π
Evaluate the line integral.∫C xy dx + y2 dy + yz dz,C is the line segment from (1, 0, -1), to (3, 4, 2)
Evaluate the line integral.∫C √xy dx + ey dy + xz dz,C is given by r(t) = t4 i + t2 j + t3 k, 0 < t < 1
Evaluate the line integral.∫C y3 dx + x2 dy, C is the arc of the parabola x = 1 - y2from (0, -1) to (0, 1)
Evaluate the line integral.∫C y dx + (x + y2)dy, C is the ellipse 4x2 + 9y2 = 36 with counterclockwise orientation
Evaluate the line integral.∫C yz cos x ds,C: x = t, y = 3 cos t, z = 3 sin t, 0 < t < π
Evaluate the line integral.∫C x ds,C is the arc of the parabola y = x2 from (0, 0) to (1, 1)
A vector field F, a curve C, and a point P are shown.(a) Is ∫C F • dr positive, negative, or zero? Explain.(b) Is div F(P) positive, negative, or zero? Explain. УА х P R RR
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The area of the region bounded by the
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.There is a vector field F such that curl F =
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If S is a sphere and F is a constant vector
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F and G are vector fields, thencurl(F •
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F and G are vector fields, thencurl(F + G)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The work done by a conservative force field
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F and G are vector fields and divF = divG,
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. S-c f(x, y) ds = -Sc f(x, y) ds %3D
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F = P i + Q j and Py = Qx in an open
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has continuous partial derivatives on R3
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has continuous partial derivatives of
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F is a vector field, then curl F is a
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F is a vector field, then div F is a
A solid occupies a region E with surface S and is immersed in a liquid with constant density p. We set up a coordinate system so that the xy-plane coincides with the surface of the liquid, and
Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives.Prove thatThese surface and triple integrals of vector functions are
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial
Use the Divergence Theorem to evaluate∫∫s (2x + 2y + z2) dSwhere S is the sphere x2 + y2 + z2 = 1.
Verify that div E = 0 for the electric field E(x) = єQ/|x|3 x.
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) = (x2, y2)
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, x2 + y2)
(a) Are the points P1 and P2 sources or sinks for the vector field F shown in the figure? Give an explanation based solely on the picture.(b) Given that F(x, y) = (x, y2), use the definition of
A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and at P2. -2 -2 2. 2.
Let F(x, y, z) = z tan-1(y2 ) i + z3 ln(x2 + 1) j + z k. Find the flux of F across the part of the paraboloid x2 + y2 + z = 2 that lies above the plane z = 1 and is oriented upward.
Use the Divergence Theorem to evaluate ʃʃS F π dS, where F(x, y, z) = z2x i + (1/3y3 + tan z) j + (x2z + y2 ) k and S is the top half of the sphere x2 + y2 + z2 = 1.That S is not a closed surface.
Use a computer algebra system to plot the vector field F(x, y, z) = sin x cos2y i + sin3y cos4z j + sin5z cos6x k in the cube cut from the first octant by the planes x = πy2, y = πy2, and z = πy2.
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = ey tan z i + y√3 - x2 j + x sin y k, S is the surface of the
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F = |r|2 r, where r = x i + y j + z k, S is the sphere with radius R and center
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F = |r| r, where r = x i + y j + z k, S consists of the hemisphere z = √1 - x2
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = (xy + 2xz) i + (x2 + y2) j + (xy - z2) k, S is the surface of the
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = (2x3 + y3) i + (y3 + z3) j + 3y2z k, S is the surface of the solid
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = z i + y j + zx k,S is the surface of the tetrahedron enclosed by
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = xey i + (z - ey)j - xy k, S is the ellipsoid x2 + 2y2 + 3z2 = 4
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = (x3 + y3) i + (y3 + z3) j + (z3 + x3) k, S is the sphere with
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = 3xy2 i + xez j + z3 k, S is the surface of the solid bounded by the
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = x2yz i + xy2z j + xyz2 k, S is the surface of the box enclosed by
Use the Divergence Theorem to calculate the surface integral ʃʃS F • dS; that is, calculate the flux of F across S.F(x, y, z) = xyez i + xy2z3 j - yez k, S is the surface of the box bounded by
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = (x2, 2y, z), E is the solid cylinder y2 + z2 < 9, 0 < x < 2
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = (z, y, x), E is the solid ball x2 + y2 + z2 < 16
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = y2z3 i + 2yz j + 4z - k, E is the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9
Verify that the Divergence Theorem is true for the vector field F on the region E.F(x, y, z) = 3x i + xy j + 2xz k, E is the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1
Suppose S and C satisfy the hypotheses of Stokes’ Theorem and f, t have continuous second-order partial derivatives. Use Exercises 24 and 26 in Section 16.5 to show the following.(a)(b)(c) Se(S Vg)
If S is a sphere and F satisfies the hypotheses of Stokes’ Theorem, show that ʃʃS curl F • dS = 0.
EvaluateʃC(y + sin x) dx + (z2 + cos y) dy + x2 dzwhere C is the curve r(t) = (sin t, cos t, sin 2t), 0 < t < 2π,. Observe that C lies on the surface z = 2xy.]
A particle moves along line segments from the origin to the points (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the origin under the influence of the force fieldF(x, y, z) = z2 i + 2xy j + 4y2 kFind
Let C be a simple closed smooth curve that lies in the plane x + y + z = 1. Show that the line integralʃC z dx - 2x dy + 3ydzdepends only on the area of the region enclosed by C and not on the
Verify that Stokes’ Theorem is true for the given vector field F and surface S.F(x, y, z) = y i + z j + x k, S is the hemisphere x2 + y2 + z2 = 1, y > 0, oriented in the direction of the
Verify that Stokes’ Theorem is true for the given vector field F and surface S.F(x, y, z) = -2yz i + y j + 3x k, S is the part of the paraboloid z = 5 - x2 - y2 that lies above the plane z = 1,
Verify that Stokes’ Theorem is true for the given vector field F and surface S.F(x, y, z) = 2y i + x j - 2 k,S is the cone z2 = x2 + y2, 0 < z < 4, oriented downward
(a) Use Stokes’ Theorem to evaluate ʃC F • dr, where F(x, y, z) = x2y i + 1/3 x3 j + xy k and C is the curve of intersection of the hyperbolic paraboloid z = y2 - x2 and the cylinder
(a) Use Stokes’ Theorem to evaluate ʃC F • dr, where F(x, y, z) = x2z i + xy2 j + z2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9, oriented
Use Stokes’ Theorem to evaluate ʃC F • dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = 2y i + xz j + (x + y) k, C is the curve of intersection of the plane z =
Use Stokes’ Theorem to evaluate ʃC F • dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = xy i + yz j + zx k, C is the boundary of the part of the paraboloid z = 1
Use Stokes’ Theorem to evaluate ʃC F • dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = i + (x + yz) j + (xy - √z) k, C is the boundary of the part of the
Use Stokes’ Theorem to evaluate ʃC F • dr. In each case C is oriented counterclockwise as viewed from above.F(x, y, z) = (x + y2) i + (y + z2) j + (z + x2) k, C is the triangle with vertices (1,
Use Stokes’ Theorem to evaluate ʃʃS curl F • dS.F(x, y, z) = exy i + exz j + x2z k, S is the half of the ellipsoid 4x2 + y2 + 4z2 = 4 that lies to the right of the xz-plane, oriented in the

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