Use the Divergence Theorem to calculate the surface integral ∫∫S 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 - y2 and the disk x2 + y2 < 1 in the xy-plane
> If f and t are twice differentiable functions, show that ∇2(f g) = f ∇2g + g ∇2 f + 2 ∇f ∙ ∇g
> If C is any piecewise-smooth simple closed plane curve and f and t are differentiable functions, show that ∫C f (x) dx + g (y) dy = 0
> A spring with a mass of 2 kg has damping constant 16, and a force of 12.8 N keeps the spring stretched 0.2 m beyond its natural length. Find the position of the mass at time t if it starts at the equilibrium position with a velocity of 2.4 m/s.
> A series circuit contains a resistor with R = 40 V, an inductor with L = 2 H, a capacitor with C = 0.0025 F, and a 12-V battery. The initial charge is Q = 0.01 C and the initial current is 0. Find the charge at time t.
> Use power series to solve the differential equation y'' – xy' - 2y = 0
> Use power series to solve the initial-value problem y'' + xy' + y = 0 y (0) = 0 y'(0) = 1
> Solve the boundary-value problem, if possible. y'' + 4y' + 29y = 0, y (0) = 1, y (π) = -e-2 π
> (a). Write the definition of the line integral of a scalar function f along a smooth curve C with respect to arc length. (b). How do you evaluate such a line integral? (c). Write expressions for the mass and center of mass of a thin wire shaped like a cu
> Solve the boundary-value problem, if possible. y'' + 4y' + 29y = 0, y (0) = 1, y (π) = -1
> Solve the initial-value problem. 9y'' + y = 3x + e-x, y (0) = 1, y'(0) = 2
> Solve the initial-value problem. y'' - 5y' + 4y = 0, y (0) = 0, y'(0) = 1
> Solve the initial-value problem. y'' - 6y' + 25y = 0, y (0) = 2, y'(0) = 1
> Solve the initial-value problem. y'' + 6y' = 0, y (1) = 3, y'(1) = 12
> Solve the differential equation. d2y/dx2 + y = csc x, 0 < x < π/2
> Solve the differential equation. d2y/dx2 – dy/dx - 6y = 1 + e-2x
> Solve the differential equation. d2y/dx2 + 4y = sin 2x
> Solve the differential equation. d2y/dx2 - 2 dy/dx + y = x cos x
> Solve the differential equation. d2y/dx2 + dy/dx - 2y = x2
> What is a vector field? Give three examples that have physical meaning.
> Prove the following identity: ∇ (F ∙ G) = (F ∙ ∇) G + (G ∙ ∇) F + F × curl G + G × curl F
> Solve the differential equation. y'' + 8y' + 16y = 0
> Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector n = 〈a, b, c〉 and has positive orientation with respect to n. Show that the plane area enclosed by C is } 6г — су)
> Evaluate the line integral. ∫C x ds, C is the arc of the parabola y = x2 from (0, 0) to (1, 1)
> Solve the differential equation. 4y'' - y = 0
> Solve the differential equation. d2y/dx2 - 4 dy/dx + 5y = e2x
> Solve the differential equation. y'' + 3y = 0
> Write expressions for the area enclosed by a curve C in terms of line integrals around C.
> State Green’s Theorem.
> Solve the differential equation. 3y'' + 4y' - 3y = 0
> State the Fundamental Theorem for Line Integrals.
> The figure depicts the sequence of events in each cylinder of a four-cylinder internal combustion engine. Each piston moves up and down and is connected by a pivoted arm to a rotating crankshaft. Let P (t) and V (t) be the pressure and volume within a cy
> Solve the differential equation. Y'' – y' - 6y = 0
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the ellipsoid x2 + 2y2 + 3z2 = 4 F(х, у, 2) — хе'і + (г — е')j — хуk,
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the sphere with center the origin and radius 2 F(x, y, z) = (x³ + y³) i+ (y' + z')j + (z²
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the cylinder y2 + z2 = 1 and the planes x = -1 and x = 2
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, wh
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the box bounded by the coordinate planes and the planes x = 3, y = 2, and z = 1
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = tan-1(x2/z2) i + x2y j + x2z2 k, S is the cone x = √y2 + z2, 0 < x < 2, oriented in the direction of the positive x-axis
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = zey i + x cos y j + xz sin y k, S is the hemisphere x2 + y2 + z2 = 16, y > 0, oriented in the direction of the positive y-axis
> 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 derivatives. || (S Vg) • n dS || (SV°g+ Vf• Vg) đV E
> 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 derivatives. V(E) = || F· dS, where F(x, y, z) = xi+ yj+ z k
> (a). Define the line integral of a vector field F along a smooth curve C given by a vector function r (t). (b). If F is a force field, what does this line integral represent? (c). If F =〈P, Q, R〉, what is the connection between the line integral of F and
> Use the Divergence Theorem to evaluate ∫∫S (2x + 2y + z2) dS where S is the sphere x2 + y2 + z2 = 1.
> Verify that div E = 0 for the electric field E (x) = ∈Q |x |3 x.
> 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. Exercises 24: Prove the identity, assuming that the ap
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = x3yz2 j + y4z3 k
> 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 tttt / 2. ! 11 !
> Evaluate the surface integral. ∫∫S (x + y + z) dS, S is the part of the half-cylinder x2 + z2 = 1, z > 0, that lies between the planes y = 0 and y = 2
> Use the Divergence Theorem to evaluate ∫∫S F ∙ dS, where F (x, y, z) = z2x i + (1/3 y3 + tan z) j + (x2z + y2) k and S is the top half of the sphere x2 + y2 + z2 = 1. [Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, wh
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the cylinder x2 + y2 = 4 and the planes z = y - 2 and z =
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the solid bounded by the paraboloid z = 1 - x2 - y2 and the xy-plane F(x, y
> (a). What is a conservative vector field? (b). What is a potential function?
> Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the tetrahedron enclosed by the coordinate planes and the plane x/a + y/b + z/c
> 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
> Evaluate the surface integral. ∫∫S x2yz dS, S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0, 3g] × [0, 2]
> Use Stokes’ Theorem to evaluate ∫C 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 plane 3x + 2y + z = 1 in the first octant
> Use Stokes’ Theorem to evaluate ∫C 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, 0, 0), (0, 1, 0), and (0, 0, 1)
> Use Stokes’ Theorem to evaluate ∫∫S 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 direction of the positive y-axis
> Evaluate the surface integral. ∫∫S (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 < u < 2, 0 < v < 1
> Let r = x i + y j + z k and r = |r |. Verify each identity. (a). ∇r = r/r (b). ∇ × r = 0 (c). ∇ (1/r) = -r/r3 (d). = ln r = r/r2
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + 5 k, S is the boundary
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + 2y j + 3z k, S is the cube wi
> In what ways are the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem similar?
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = yz i + zx j + xy k, S is the surfac
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = y j - z k, S consists of the parabo
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = y i - x j + 2z k, S is the hemisphe
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + z2 k, S is the sphere w
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = -x i - y j + z3 k, S is the part of
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = xy i + yz j + zx k, S is the part o
> Evaluate the surface integral ∫∫S F ∙ dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = z i + y j + x k, S is the helicoid
> Evaluate the surface integral. ∫∫S xz dS, S is the boundary of the region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5
> Evaluate the surface integral. ∫∫S (x2z + y2z) dS, S is the hemisphere x2 + y2 + z2 = 4, z > 0
> Evaluate the surface integral. ∫∫S y2 dS, S is the part of the sphere x2 + y2 + z2 = 1 that lies above the cone z = √x2 + y2
> State the Divergence Theorem.
> 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 positive y-axis
> Evaluate the surface integral. ∫∫S y2z2 dS, S is the part of the cone y = √x2 + z2 given by 0 < y < 5
> Evaluate the surface integral. ∫∫S z2 dS, S is the part of the paraboloid x = y2 + z2 given by 0 < x < 1
> Evaluate the surface integral. ∫∫S y dS, S is the surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
> Evaluate the surface integral. ∫∫S x dS, S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)
> Evaluate the surface integral. ∫∫S xz dS, S is the part of the plane 2x + 2y + z = 4 that lies in the first octant
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = xy2z2 i + x2yz2 j + x2y2z k
> Evaluate the surface integral. ∫∫S (x2 + y2) dS, S is the surface with vector equation r (u, v) =〈2uv, u2 - v2, u2 + v2〉, u2 + v2 < 1
> Evaluate the surface integral. ∫∫S y dS, S is the helicoid with vector equation r (u, v) =〈u cos v, u sin v, v〉, 0 < u < 1, 0 < v < π
> Evaluate the surface integral. ∫∫S xyz dS, S is the cone with parametric equations x = u cos v, y = u sin v, z = u, 0 < u < 1, 0 < v < π/2
> State Stokes’ Theorem.
> Suppose that f (x, y, z) = g (√x2 + y2 + z2), where t is a function of one variable such that g (2) = 25. Evaluate ∫∫S f (x, y, z) dS, where S is the sphere x2 + y2 + z2 = 4.
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = xyez i + yzex k
> Find a parametric representation for the surface. The plane that passes through the point (0, -1, 5) and contains the vectors 〈2, 1, 4〉 and 〈-3, 2, 5〉
> Evaluate the surface integral. ∫∫S x dS, S is the surface y = x2 + 4z, 0 < x < 1, 0 < z < 1
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C y3 dx - x3 dy, C is the circle x2 + y2 = 4
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = 〈arctan (xy), arctan (yz), arctan (zx)〉
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = 〈ex sin y, ey sin z, ez sin x〉
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = ln (2y + 3z) i + ln (x + 3z) j + ln (x + 2y) k
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = √x/ 1 + z i + √y 1 + x j + √z 1 + y k
> Find the area of the surface. The surface with parametric equations x = u2, y = uv, z = }v², 0 < u < 1,0 < v< 2
> If F = P i + Q j, how do you determine whether F is conservative? What if F is a vector field on R3?
> Find the area of the surface. The helicoid (or spiral ramp) with vector equation r(u, v) = u cos v i + u sin v j + v k, 0 <u < 1,
> (a). Show that a constant force field does zero work on a particle that moves once uniformly around the circle x2 + y2 = 1. (b). Is this also true for a force field F (x) = kx, where k is a constant and x = 〈x, y〉?
> Suppose there is a hole in the can of paint in Exercise 45 and 9 lb of paint leaks steadily out of the can during the man’s ascent. How much work is done? Exercise 45: A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a si
> Find the area of the surface. The part of the surface z = xy that lies within the cylinder x2 + y2 = 1
