Use a calculator to evaluate the line integral correct to four decimal places. ∫C F ∙ dr, where F (x, y, z) = yzex i + zxey j + xyez k and r(t) = sin t i + cos t j + tan t k, 0 < t < π/4
> 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
> Find the area of the surface. The part of the surface z = 4 - 2x2 + y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 1)
> Find the area of the surface. The surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
> Find the area of the surface. The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2 + y2 = 3
> Find the area of the surface. The part of the plane with vector equation r(и, v) — (и + 0, 2 — Зи, 1 + и — в) that is given by 0 <u < 2, –1<v<1
> Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = sin yz i + sin zx j + sin xy k
> Find the area of the surface. The part of the plane 3x + 2y + z = 6 that lies in the first octant
> 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
> Find an equation of the tangent plane to the given parametric surface at the specified point. r(u, v) = sin u i + cos u sin vj + sin v k; u = T/6, v = /6 %3D
> Find an equation of the tangent plane to the given parametric surface at the specified point. r(u, v) = u cos vi + u sin vj + v k; u = 1, v = "/3
> Find an equation of the tangent plane to the given parametric surface at the specified point. r(u, v) = u cos vi + u sin vj + v k; u= 1, v = T/3 %3D
> Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity: where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and t exist and a
> Find parametric equations for the surface obtained by rotating the curve x = 1/y, y > 1, about the y-axis and use them to graph the surface.
> Find parametric equations for the surface obtained by rotating the curve y = 1/ (1 + x2), -2 < x < 2, about the x-axis and use them to graph the surface.
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (∇f &Atild
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (F × G) = G &ac
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by curl) (f F) = f curl F + (â
> Find a parametric representation for the surface. The part of the sphere x2 + y2 + z2 = 36 that lies between the planes z = 0 and z = 3√3
> Evaluate the line integral. ∫C F ∙ dr, where F (x, y) = xy i + x2 j and C is given by r (t) = sin t i + (1 + t) j, 0 < t < π
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by curl (F + G) = curl F + curl G
> Find a parametric representation for the surface. The part of the sphere x2 + y2 + z2 = 4 that lies above the cone z = √x2 + y2
> Show that any vector field of the form F (x, y, z) = f (y, z) i + g (x, z) j + h (x, y) k is incompressible.
> Find a parametric representation for the surface. The part of the hyperboloid 4x2 - 4y2 - z2 = 4 that lies in front of the yz-plane
> Find a parametric representation for the surface. The plane through the origin that contains the vectors i - j and j - k
> A particle starts at the origin, moves along the x-axis to (5, 0), then along the quarter-circle x2 + y2 = 25, x > 0, y > 0 to the point (0, 5, and then down the y-axis back to the origin. Use Green’s Theorem to find the work done
> Use Green’s Theorem to find the work done by the force F (x, y) = x (x + y) i + xy2 j in moving a particle from the origin along the x-axis to (1, 0), then along the line segment to (0, 1), and then back to the origin along the y-axis.
> Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral. P (x, y) = 2x - x3y5, Q (x, y) = x3y8, C is the ellipse 4x2 + y2 = 4
> 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) = z cos y i + xz sin y j + x cos y k
> 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) = xyz4 i + x2z4 j + 4x2yz3 k
> Evaluate the line integral. ∫C xy dx + y2 dy + yz dz, C is the line segment from (1, 0, -1), to (3, 4, 2)
> Use Green’s Theorem to evaluate ∫C F ∙ dr. (Check the orientation of the curve before applying the theorem.) F (x, y) =〈y - cos y, x sin y〉, C is the circle (x – 3)2 + (y + 4)2 = 4 oriented clockwise
> Use Green’s Theorem to evaluate ∫C F ∙ dr. (Check the orientation of the curve before applying the theorem.) F (x, y) = 〈e^(-x)+ y^2,e^(-y) x^2 〉, C consists of the arc of the curve y = cos x from (-π/2, 0) to (π/2, 0) and the line segment from (π/2, 0
> Use Green’s Theorem to evaluate ∫C F ∙ dr. (Check the orientation of the curve before applying the theorem.) F (x, y) =〈y cos x - xy sin x, xy + x cos x〉, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C (1 - y3) dx + (x3 + ey 2) dy, C is the boundary of the region between the circles x2 + y2 = 4 and x2 + y2 = 9
> Evaluate the line integral ∫C F ∙ dr, where C is given by the vector function r(t). F (x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2 i + t3 j - 2t k, 0 < t < 2
> Evaluate the line integral, where C is the given curve. ∫C (y + z) dx + (x + z) dy + (x + y) dz, C consists of line segments from (0, 0, 0) to (1, 0, 1) and from (1, 0, 1) to (0, 1, 2)
> Evaluate the line integral, where C is the given curve. ∫C z2 dx + x2 dy + y2 dz, C is the line segment from (1, 0, 0) to (4, 1, 2)
> Evaluate the line integral, where C is the given curve. ∫Cy dx + z dy + x dz, C: x = √t, y = t, z = t2, 1 < t < 4
> Evaluate the line integral, where C is the given curve. ∫C xyeyz dy, C: x = t, y = t2, z = t3, 0 < t < 1
> 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
> How do you use power series to solve a differential equation?
> Evaluate the line integral, where C is the given curve. ∫C (x2 + y2 + z2) ds, C: x = t, y = cos 2t, z = sin 2t, 0 < t < π2
> Evaluate the line integral, where C is the given curve. ∫C xeyz ds, C is the line segment from (0, 0, 0) to (1, 2, 3)
> Evaluate the line integral, where C is the given curve. ∫C y2z ds, C is the line segment from (3, 1, 2) to (1, 2, 5)
> If C is a smooth curve given by a vector function r (t), a (v• dr = v · [r(b) – r(a)]
> Evaluate the line integral, where C is the given curve. ∫C y ds, C: x − t2, y = 2t, 0 < t < 3
> A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gr
> An object with mass m moves with position function r(t) = a sin t i + b cos t j + ct k, 0 < t < y2. Find the work done on the object during this time period.
> The position of an object with mass m at time t is r(t) = at2 i + bt3 j, 0 < t < 1. (a). What is the force acting on the object at time t? (b). What is the work done by the force during the time interval 0 < t < 1?
> The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = 〈x, y, z〉 is F (r) = Kr/|r |3 where K is a constant. (See Example 16.1.5.) Find the work done as the particle moves along a straight
> Find the work done by the force field F (x, y, z) = 〈x - y^2, y - z^2, z - x^2 〉 on a particle that moves along the line segment from (0, 0, 1) to (2, 1, 0).
> Evaluate the line integral. ∫C y3 dx + x2 dy, C is the arc of the parabola x = 1 - y2 from (0, -1) to (0, 1)
> Find the work done by the force field F (x, y) = x2 i + yex j on a particle that moves along the parabola x = y2 + 1 from (1, 0) to (2, 1).
> We have seen that all vector fields of the form F = ∇g satisfy the equation curl F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the questi
> If a wire with linear density ρ (x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined as Find the moments of inertia for the wire in Exercise 35. Exercise 35: (a). Write the formulas similar
> This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w − k, where is the angular speed of B, that is, the tangential
> Use Green’s first identity to show that if f is harmonic on D, and if f (x, y) = 0 on the boundary curve C, then ∫∫D |∇f |2 dA = 0. (Assume the same hypotheses as in Exercise 33.)
> Recall from Section 14.3 that a function t is called harmonic on D if it satisfies Laplace’s equation, that is, ∇2g = 0 on D. Use Green’s first identity (with the same hypotheses as in Exercise 33) to
> Use Green’s first identity (Exercise 33) to prove Green’s second identity: first identity: where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and
> Let r = x i + y j + z k and r = |r |. If F = r/rp, find div F. Is there a value of p for which div F = 0?
> Let r = x i + y j + z k and r = |r |. Verify each identity. (a). = r = 3 (b). = ∙ (r r) = 4r (c). ∇2r3 = 12r
> Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then aP aR aR ду ax az ax dz ду
> Evaluate the line integral. ∫C y dx + (x + y2) dy, C is the ellipse 4x2 + 9y2 = 36 with counterclockwise orientation
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (f F) = f div F + F â&
> Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then f F, F ∙ G, and F × G are defined by div (F + G) = div F + div G
> Show that any vector field of the form F (x, y, z) = f (x) i + g (y) j + h (z) k where f, t, h are differentiable functions, is irrotational.
> Is there a vector field G on R3 such that curl G =〈x, y, z〉? Explain.
> Is there a vector field G on R3 such that curl G =〈x sin y, cos y, z - xy〉? Explain.
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C y4 dx + 2xy3 dy, C is the ellipse x2 + 2y2 = 2
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C (y + e^√x) dx + (2x + cos y2) dy, C is the boundary of the region enclosed by the parabolas y = x2 and x = y2
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C (x2 + y2) dx + (x2 - y2) dy, C is the triangle with vertices (0, 0), (2, 1), and (0, 1)
> Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C x2y2 dx 1 xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)
> Evaluate the line integral. ∫C yz cos x ds, C: x = t, y = 3 cos t, z = 3 sin t, 0 < t < π
> Find the work done by the force field F (x, y) = x i + (y + 2) j in moving an object along an arch of the cycloid r(t) = (t - sin t) i + (1 - cos t) j 0 < t < 2 π
> If a wire with linear density ρ (x, y) lies along a plane curve C, its moments of inertia about the x- and y-axes are defined as Find the moments of inertia for the wire in Example 3. -[v°ptx, y) ds I, = x°p(x, y) ds
> Find the mass and center of mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 < t < 2 π, if the density at any point is equal to the square of the distance from the origin.
> (a). Write the formulas similar to Equations 4 for the center of mass (z ̅, y ̅, z ̅) of a thin wire in the shape of a space curve C if the wire has density function ρ (x, y, z). (b). Find the center of mass of a wire in the shape of the helix x = 2 sin
> A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is ρ (x, y) = kxy, find the mass and center of mass of the wire.
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = 0.3 i - 0.4 j ZA
> Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices. f (x, y) = sin √ (x^2 + y^2) IV 4 -4 4 سر ه م -4
> What does Clairaut’s Theorem say?
> Complete the proof of the special case of Green’s Theorem by proving Equation 3.
> Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C xy dx + x2y3 dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2)
> If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl (F × G) = F div G - G div F + (G ∙ ∇) F – (F ∙∇) G
