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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) (u sin u) cos vi + (1 - cos u) sin vj + uk = 0 ≤u≤ π, 0 ≤ y ≤ 2π -
Evaluate S: r(u, v) = ui + vj + 2vk, 0 ≤ u ≤ 1, 0 ≤ v ≤ SS f (x, y) ds.
Use Green’s Theorem to evaluate the line integral. So C C: x² + y² = a² ex cos 2y dx - 2e* sin 2y dy
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = (cos y + y cos x)i + (sin x - x sin y)j + xyzk
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y) = x² + 2y²
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z)= yzi + (2 − 3y)j + (x² + y²)k, x² + y² ≤ 16 S: the first-octant
(a) Find a parametrization of the path C(b) Evaluatealong C.C: y-axis from y = 1 to y = 9 √√(x + 4 + 4√y) ds C
Use Green’s Theorem to evaluate the line integral. So 2 arctandx + In(x² + y²) dy X C: x = 4 + 2 cos 0, y = 4 + sin 0
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)(a)(b)(c)(d) C F. dr.
Evaluate S: r(u, v) = ui + vj + 2vk, 0 ≤ u ≤ 1, 0 ≤ v ≤ SS f (x, y) ds.
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = cos³ u cos vi+ sin³ u sin vj + uk 0 su≤772₁2 0 ≤ y ≤ 2π
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = (3x - y)i + (y – 2z)j + (z − 3x)k
Determine how thegraph of the surface s(u, v) differs from the graph of r(u, v) =u cos vi + u sin vj + u²k (see figure), where 0 ≤ u ≤ 2 and0 ≤ y ≤ 2π. (It is not necessary to graph s.)
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y) = x2 - 1/4y2
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = xyzi + yj + zk S: z = x², 0≤x≤ a, 0≤ y ≤ a N is the downward unit
Evaluate the following SS f(x, y, z) dS. s.
Use Green’s Theorem to evaluate the line integral. cos y dx + (xy - x sin y) dy Jc C: boundary of the region lying between the graphs of y = x and y = √√√x
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)(a)(b) C: parabola y = 4 - x² from (2, 0) to (0,4) C F. dr.
(a) Find a parametrization of the path C(b) Evaluatealong C.C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1) √√(x + 4 + 4√y) ds C
Evaluatewhere S is the closed surface of the solid bounded by the graphs of x = 4 and z = 9 - y², and the coordinate planes.F(x, y, z) = (4xy + z²)i + (2x² + 6yz)j + 2xzk JSJ curl F. NdS
Determine how the graph of the surface s(u, v) differs from the graph of r(u, v) = u cos vi + u sin vj + u²k (see figure), where 0 ≤ u ≤ 2 and 0 ≤ y ≤ 2π. (It is not necessary to graph
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = arcsin xi + xy2j + yz2k
Find the conservative vector field for the potential function by finding its gradient.g(x, y) = 5x² + 3xy + y² +
(a) Find a parametrization of the path C(b) Evaluatealong C.C: counterclockwise around the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) √√(x + 4 + 4√y) ds C
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = xyzi + yj + zk, x² + y² ≤ a² S: the first-octant portion of z = x²
Evaluatewhere S is the closed surface of the solid bounded by the graphs of x = 4 and z = 9 - y², and the coordinate planes.F(x, y, z) = xy cos zi + yz sin xj + xyzk JSJ curl F. NdS
Evaluate the following SS f(x, y, z) dS. s.
Use Green’s Theorem to evaluate the line integral. So (e-x²/2 - y) dx + (e-y²/2 + x) dy C: boundary of the region lying between the graphs of the circle x = 6 cos 0, y = 6 sin 0 and the ellipse x
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = (x² - y)i — (x + sin²y) j
Find the conservative vector field for the potential function by finding its gradient.g(x, y) = sin 3x cos 4y
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)(a) r₁(t) = t³i + t²j, 0 ≤ t ≤ 2(b) r₂(t) = 2 cos ti + 2 sin tj, 0 ≤ t ≤
Determine how the graph of the surface s(u, v) differs from the graph of r(u, v) = u cos vi + u sin vj + u²k (see figure), where 0 ≤ u ≤ 2 and 0 ≤ y ≤ 2π. (It is not necessary to graph
(a) Find a piecewise smooth parametrization of the path C shown in the figure, and (b) Evaluatealong C. Jc (2x + y²z) ds
Use Green’s Theorem to evaluate the line integral. Se (x - 3y) dx + (x + y) dy C: boundary of the region lying between the graphs of x² + y² = 1 and x² + y² = 9
Evaluate the following SS f(x, y, z) dS. s.
Find (a) The divergence of the vector field F (b) The curl of the vector field F Z F(x, y, z) = ² i + j + z²k X
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = ln(x² + y²)i + ln(x² + y²)j + zk
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = yzi + xzj + xyk(a) r1(t) = ti + 2j + tk, 0 ≤ t ≤ 4(b) r₂(t) = t²i +
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y, z) = 6xyz
Determine how the graph of the surface s(u, v) differs from the graph of r(u, v) = u cos vi + u sin vj + u²k (see figure), where 0 ≤ u ≤ 2 and 0 ≤ y ≤ 2π. (It is not necessary to graph s.)
The motion of aliquid in a cylindrical container of radius 1 is described by thevelocity field F(x, y, z). Find ∫s∫ (curl F). NdS, where S is the upper surface of the cylindrical container.F(x,
Use Green’s Theorem to evaluate the line integral. So 3x²e dx e dy C: boundary of the region lying between the squares with vertices (1, 1), (-1, 1), (-1, -1), and (1, -1), and (2, 2), (-2, 2),
Evaluate the follwoing SS f(x, y, z) dS. s.
(a) Find a piecewise smooth parametrization of the path C shown in the figure, and (b) Evaluatealong C. Jc (2x + y²z) ds
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y, z) = √x² + 4y² + 2²
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = i + zj + yk(a) r₁(t) = cos ti + sin tj + t²k, 0 ≤ t ≤ π(b) r₂(t)
The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field F(x, y, z). Find ∫s∫ (curl F). NdS, where S is the upper surface of the cylindrical container.F(x,
Find the total mass of two turns of a spring with density ρ in the shape of the circular helixρ(x, y, z) = 1/2(x² + y² + z²) r(t) = 2 cos ti + 2 sin tj + tk, 0 ≤ t ≤ 4m.
Evaluate the line integral along the given path(s).(a) C: line segment from (0, 0) to (3, 4)(b) C: x² + y² = 1, one revolution counterclockwise,starting at (1, 0) So (x² + y²) ds
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = (2y + x)i + (x² − z)j + (2y - 4z)k (a) r1(t) = ti + t²j+ k, 0 ≤
Evaluate the follwoing SS f(x, y, z) dS. s.
How do you determine whether a point (x0, Y0, Z0) in a vector field is asource, a sink, or incompressible?
Find a vector-valued function whose graph is the indicated surface.The plane z = y
Find the conservative vector field for the potential function by finding its gradient.g(x, y, z) = z + yex²
Verify thatfor any closed surface S. SS Is. curl F N dS = 0 .
Evaluate the follwoing SS f(x, y, z) dS. s.
Find the total mass of two turns of a spring with density ρ in the shape of the circular helixρ(x, y, z) = z r(t) = 2 cos ti + 2 sin tj + tk, 0 ≤ t ≤ 4m.
Find the conservative vector field for the potential function by finding its gradient. g(x, y, z) = y Y Z + X N XZ y
Evaluate the line integral along the given path(s).(a) C: line segment from (0, 0) to (5, 4)(b) C: counterclockwise(0, 0), (4, 0), and (0, 2)around the triangle with vertices с xy ds
Use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.F(x, y) = xyi + (x + y)jC: x² + y² = 1
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = -yi + xj + 3xz²k(a) r₁(t) = cos ti + sin tj + tk, 0 ≤ t ≤ π(b)
The graph of a vector field F is shown. Does the graph suggest that the divergence of F at P is positive, negative, or zero? 2 //-2- P. X
Find a vector-valued function whose graph is the indicated surface.The plane x + y + z = 6
Evaluate the line integral along the given path(s). [0x²+3 C: r(t) = (1 sin t)i + (1 - cost)j, 0≤ t ≤ 2π (x² + y²) ds
Give a physical interpretation of curl.
Let C be a constant vector. Let S be an oriented surface with a unit normal vector N, bounded by a smooth curve C. Prove that [[ c. Nds 1 [(Cxr). = 2 dr.
Find the flux of F through S,where N is the upward unit normal vector to S. JsJ F. NdS
Find the conservative vector field for the potential function by finding its gradient.h(x, y, z) = xy In(x + y)
Find a vector-valued function whose graph is the indicated surface.The cone y = √4x² + 9z²
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = ez(yi + xj + xyk)(a) r1(t) = 4 cos ti + 4 sin tj + 3k, 0 ≤ t ≤ π(b)
Evaluate the line integral along the given path(s). le (x² + y²) ds C: r(t) = (cost + t sin t)i + (sin t - t cos t)j, 0 ≤ t ≤ 2π
(a) Use the Divergence Theorem to verify that the volume of the solid bounded by a surface S is(b) Verify the result of part (a) for the cube bounded by x = 0, x = a, y = 0, y = a, z = 0, and z = a.
Use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.F(x, y) = (x³/2 - 3y)i + (6x + 5√y)jC: boundary of the
Let S1 be the portion of the paraboloid lying above the xy-plane, and let S₂ be the hemisphere, as shown in the figures. Both surfaces are oriented upward.For a vector field F(x, y, z) with
Find the total mass of the wire with density ρ.r(t) = cos ti + sin tj, ρ(x, y) = x + y + 2, 0 ≤ t ≤ π
For the constant vector field F(x, y, z) = a₁i + a₂j + a3k, verify the following integral for any closed surface S. LSF Js JSJ F. NdS = 0
Find a vector-valued function whose graph is the indicated surface. The cone x = √16y² + z²
Find the conservative vector field for the potential function by finding its gradient.h(x, y, z) = x arcsin yz
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y, z) = y sin zi + x sin zj + xy cos xk(a) r₁(t) = t²i + t²j, 0 ≤ t ≤ 2(b)
Find the flux of F through S,where N is the upward unit normal vector to S. JsJ F. NdS
Evaluate the line integral along the given path(s).(a) C: line segment from (0, 0) to (3, -3)(b) C: one revolution counterclockwise around the circleX = 3 cos t, y = 3 sin t So C (2x - y) dx + (x +
Use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.F(x, y) = (3x² + y)i + 4xy²jC: boundary of the region lying
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. So C: smooth curve from (0, 0) to (3, 8) (3yi + 3xj) dr
LetProve or disprove that there is a a vector-valued function F(x, y, z) = (M(x, y, z), N(x, y, z), P(x, y, z)) with the following properties:(i) M, N, P have continuous partial derivatives for
Find a vector-valued function whose graph is the indicated surface.The cylinder x² + y² = 25
Determine whether the vector field is conservative. F(x, y) = 1 (yi - xj) x²
Determine whether the vector field is conservative.F(x, y) = xy²i + x²yj
For the vector field F(x, y, z) = xi + yj + zk, verify the following integral, where V is the volume of the solid bounded by the closed surface S. SS₁ JS FN dS = 3V
Find the total mass of the wire with density ρ.r(t) = t²i + 2tj + tk, ρ(x, y, z) = kz (k > 0), 1 ≤ t ≤ 3
Evaluate the line integral along the given path(s). So (2x - y) dx + (x + 3y) dy C: r(t) = (cost + t sin t)i + (sin t t sin t) j, 0 ≤ t ≤ π/2
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. [ [2(x + y)i + 2(x + y)j] · dr C C: smooth curve from (-1, 1) to (3,
Use a line integral to find the area of the region R.R: region bounded by the graph of x² + y² = a²
Find the flux of F through S,where N is the upward unit normal vector to S. JsJ F. NdS
Find a vector-valued function whose graph is the indicated surface.The cylinder 4x² + y² = 16
Use a computer algebra system to evaluate the line integral over the given path. le r(t) = a cos³ ti + a sin³ tj, 0 ≤ t ≤ π/2 (2x + y) ds
For the vector field F(x, y, z) = xi + yj + zk, verify that |F|| S. F. NdS 3 Allfov. SSS dV. |F|| Q
Find the total mass of the wire with density ρ.r(t) = 2 cos ti + 2 sin tj + 3tk, ρ(x, y, z) = k + z(k > 0), 0 ≤ t ≤ 2π
Use a line integral to find the area of the region R.R: triangle bounded by the graphs of x = 0, 3x - 2y = 0, and x + 2y = 8

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