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 + (z – y) j + x k, S is the surface of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1)
> 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
> A spring with a mass of 2 kg has damping constant 14, and a force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1 m beyond its natural length and then released with zero velocity. Find the positi
> Use power series to solve the differential equation. y' = xy
> A spring has natural length 0.75 m and a 5-kg mass. A force of 25 N is needed to keep the spring stretched to a length of 1 m. If the spring is stretched to a length of 1.1 m and then released with velocity 0, find the position of the mass after t second
> What are the traces of a surface? How do you find them?
> (a). How do you find the distance from a point to a line? (b). How do you find the distance from a point to a plane? (c). How do you find the distance between two lines?
> (a). Describe a method for determining whether three points P, Q, and R lie on the same line. (b). Describe a method for determining whether four points P, Q, R, and S lie in the same plane.
> (a). How do you tell if two vectors are parallel? (b). How do you tell if two vectors are perpendicular? (c). How do you tell if two planes are parallel?
> Write a vector equation and a scalar equation for a plane.
> Write a vector equation, parametric equations, and symmetric equations for a line.
> 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. P
> How do you find a vector perpendicular to a plane?
> How are cross products useful?
> How are dot products useful?
> If a is a vector and c is a scalar, how is ca related to a geometrically? How do you find ca algebraically?
> A spring with an 8-kg mass is kept stretched 0.4 m beyond its natural length by a force of 32 N. The spring starts at its equilibrium position and is given an initial velocity of 1 m/s. Find the position of the mass at any time t.
> 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.
> Graph the particular solution and several other solutions. What characteristics do these solutions have in common? y'' + 3y' + 2y = cos x
> Use Stokes’ Theorem to evaluate ∫C 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 = y + 2 and the cylinder x2 + y2 = 1
> Solve the differential equation or initial-value problem using the method of undetermined coefficients. у" — у' %3 хе", У(0) — 2, у(0) — 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. r(u, v) = (u, v², –v), -2 <u < 2, -2 < v< 2
> (a). What is an oriented surface? Give an example of a non-orientable surface. (b). Define the surface integral (or flux) of a vector field F over an oriented surface S with unit normal vector n. (c). How do you evaluate such an integral if S is a parame
> Solve the differential equation or initial-value problem using the method of undetermined coefficients. у" — 2у + 5у %3Dsin x, y(0) 3D 1, у' (0) — 1
> 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
> Solve the differential equation or initial-value problem using the method of undetermined coefficients. y" – 3y' = sin 2x
> A hemisphere H and a portion P of a paraboloid are shown. Suppose F is a vector field on R3 whose components have continuous partial derivatives. Explain why [| curl F· dS || curl F ds H P P H
> Solve the initial-value problem. 4y'' - 20y' + 25y = 0, y (0) = 2, y' (0) = -3
> Solve the initial-value problem. y'' - 6y' + 10y = 0, y (0) = 2, y' (0) = 3
> (a). Use Stokes’ Theorem to evaluate ∫C 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 x2 + y2 = 1, oriented counterclockwise as viewed from above. (b). Gr
> (a). Use Stokes’ Theorem to evaluate ∫C 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 counterclockwise as viewed from above. (b). Graph both the plan
> Use Stokes’ Theorem to evaluate ∫C 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 - x2 - y2 in the first octant
> 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(и, в) — (и*, о?, и + »), -1 <u<1, - : 1 <v < 1
> (a). Write the definition of the surface integral of a scalar function f over a surface S. (b). How do you evaluate such an integral if S is a parametric surface given by a vector function r (u, v)? (c). What if S is given by an equation z = g (x, y)? (d
> Verify that the Divergence Theorem is true for the vector field F on the region E. F (x, y, z) = y2z3 i + 2yz j + 4z2 k, E is the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9
> Let F (x, y) = -y i + x j/ x2 + y2. (a). Show that ∂P/∂y = ∂Q/∂x. (b). Show that ∫C F ∙ dr is not independent of path. [Hint: 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,
> Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. {(x, y) | (x, y) ≠ (2, 3)}
> Find an equation of the tangent plane to the given parametric surface at the specified point. х — и + 0, у %3 Зи*, г — и — 0; (2, 3, 0)
> A solid occupies a region E with surface S and is immersed in a liquid with constant density . We set up a coordinate system so that the xy-plane coincides with the surface of the liquid, and positive values of z are measured downward into the liquid. Th
> 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) = x2 i + y2 j + z2 k, S is the bounda
> Use Exercise 29 to show that the line integral ∫C y dx + x dy + xyz dz is not independent of path. Exercise 29: Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivat
> Use a graphing device to produce a graph that looks like the given one. y
> Use a graphing device to produce a graph that looks like the given one. -3 -3 05 y
> Is the vector field shown in the figure conservative? Explain. y.
> (a). What is an initial-value problem for a second-order differential equation? (b). What is a boundary-value problem for such an equation?
> Find a parametric representation for the surface. The part of the ellipsoid x2 + 2y2 + 3z2 = 1 that lies to the left of the xz-plane
> Plot the vector field and guess where div F > 0 and where div F F(x, y) = (xy, x + y²)
> Let C be a simple closed smooth curve that lies in the plane x 1 y + z = 1. Show that the line integral ∫C z dx - 2x dy + 3ydz depends only on the area of the region enclosed by C and not on the shape of C or its location in the plane.
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. r(u, v) = (u³ – u) i + v² j + u
> 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, oriented upward
> (a). If C is the line segment connecting the point (x1, y1) to the point (x2, y2), show that (b). If the vertices of a polygon, in counterclockwise order, are (x1, y1), (x2, y2), . . . , (xn , yn), show that the area of the polygon is (c). Find the a
> Evaluate the surface integral. ∫∫S (x2 + y2 + z2) dS, S is the part of the cylinder x2 + y2 = 9 between the planes z = 0 and z = 2, together with its top and bottom disks
> Evaluate ∫C (y + sin x) dx + (z2 + cos y) dy + x3 dz where C is the curve r (t) =〈sin t, cos t, sin 2t〉, 0 < t < 2π . [Hint: Observe that C lies on the surface z = 2xy.]
> Identify the surface with the given vector equation. r (u, v) = u2 i + u cos v j + u sin v k
> Show that F is conservative and use this fact to evaluate ∫C F ∙ dr along the given curve. F(x, y, z) = e'i + (xe' + e*)j + ye k, Cis the line segment from (0, 2, 0) to (4, 0, 3)
> Let H be the hemisphere x2 + y2 + z2 = 50, z > 0, and suppose f is a continuous function with f (3, 4, 5) = 7, f (3, -4, 5) = 8, f -3, 4, 5) = 9, and f (-3, -4, 5) = 12. By dividing H into four patches, estimate the value of ∫∫H f (x, y, z) dS.
> Use Stokes’ Theorem to evaluate ∫∫S curl F dS. F (x, y, z) = x2 sin z i + y2 j + xy k, S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-plane, oriented upward
> Use one of the formulas in (5) to find the area under one arch of the cycloid x = t2 sin t, y = 1 - cos t.
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. x = sin u, y = cos u sin v, z=
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. х— cos'и сos'p, у%3 sin'u cos',
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. х%3 (1 — и)(3 + соs D) cos 4ти,
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. r(u, ») — ир?і + и?o ј + (u? —
> 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 constant. I II ZA III IV ZA ZA V ZA VI z. r(u, v) = u cos vi + u sin vj +
> Use a computer to graph the parametric surface. Get a printout and indicate on its which grid curves have u constant and which have v constant. x = cos u, y = sin u sin v, z= cos v, 0 <u< 27, 0<v< 27
> 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 = sin v, y = cos u sin 4v, z = sin 2u sin 4v, 0 <u< 2m, -T/2 < v< T/2
> Show that F is conservative and use this fact to evaluate ∫C F ∙ dr along the given curve. F(x, y) = (4x³y² – 2xy) i + (2x*y – 3x²y² + 4y®)j. C: r(t) = (t + sin mt) i + (2t + cos Tt)j, 0 < t< 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. r(u, v) = (u, sin(u+ v), sin v), -T <uE T, -T <V <T
> 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(и, v) — (и*, и sin v, и cos v), -1 <u < 1, 0 <o< 27
> Identify the surface with the given vector equation. r (s, t) = 〈s cos t, s sin t, s〉
> A surface S consists of the cylinder x2 + y2 = 1, -1 Estimate the value of ∫∫S f (x, y, z) dS by using a Riemann sum, taking the patches Sij to be four quarter-cylinders and the top and bottom disks. f(±1, 0, 0
> Let S be the surface of the box enclosed by the planes x = ±1, y = ±1, z = ±1. Approximate ∫∫S cos (x + 2y + 3z) dS by using a Riemann sum as in Definition 1, taking the patches Sij to be the squares that are the faces of the box S and the points P*ij to
> Maxwell’s equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows: where c is the speed of light. Use these equations to prove the fo
> Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y, z) = -y i ZA
> Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. {(x, y) | 1 < |x | < 2}
> (a). What happens to the spiral tube in Example 2 (see Figure 5) if we replace cos u by sin u and sin u by cos u? (b). What happens if we replace cos u by cos 2u and sin u by sin 2u?
> Show that F is a conservative vector field. Then find a function f such that F = ∇f. F(x, y, z) = sin yi + x cos yj – sin z k
> Show that the line integral is independent of path and evaluate the integral. ∫C sin y dx + (x cos y - sin y) dy, C is any path from (2, 0) to (1, π)
> Identify the surface with the given vector equation. r (u, v) = (u + v) i + (3 – v) j + (1 + 4u + 5v) k
> Determine whether the points P and Q lie on the given surface. r (u, v) = 〈1 + u - v, u + v2, u2 - v2〉 P (1, 2, 1), Q (2, 3, 3)
> Determine whether the points P and Q lie on the given surface. r (u, v) = 〈u + v, u - 2v, 3 + u – v〉 P (4, -5, 1), Q (0, 4, 6)
> A particle moves in a velocity field V (x, y) = 〈x^2, x, + y^2 〉 If it is at position s2, 1d at time t − 3, estimate its location at time t = 3.01.
> If you have a CAS that plots vector fields (the command is field plot in Maple and Plot Vector Field or Vector Plot in Mathematica), use it to plot F (x, y) = (y2 - 2xy) i +) (3xy - 6x2) j Explain the appearance by finding the set of points (x, y) such t
> Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices. f (x, y) = (x + y)2 II 4 -4 4 -4
> 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) = cos x - 2 sin y
> Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices. F (x, y, z) = x i + y j + z k IV z 0 z 0- -10 y 10-1
> (a). What is a function of two variables? (b). Describe three methods for visualizing a function of two variables.
> (a). What is a parametric surface? What are its grid curves? (b). Write an expression for the area of a parametric surface. (c). What is the area of a surface given by an equation z = t (x, y)?
> (a). Find the work done by the force field F (x, y) = x2 i + xy j on a particle that moves once around the circle x2 + y2 = 4 oriented in the counterclockwise direction. (b). Use a computer algebra system to graph the force field and circle on the same s
> (a). Evaluate the line integral ∫C F ∙ dr, where F (x, y) = ex-1 i + xy j and C is given by r(t) = t2 i + t3 j, 0 (b). Illustrate part (a) by using a graphing calculator or computer to graph C and the vectors from the
> Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. {(x, y) | 1 < x2 + y2 < 4, y > 0}
> Use Green’s Theorem to prove the change of variables formula for a double integral (Formula 15.9.9) for the case where f (x, y) = 1: Here R is the region in the xy-plane that corresponds to the region S in the uv-plane under the trans
> Let F = ∇f, where f (x, y) = sin (x - 2y). Find curves C1 and C2 that are not closed and satisfy the equation. (a) f. F· dr = 0 JCI (b) f. (b) F. dr = 1
> Use the method of Example 5 to calculate ∫C F ∙ dr, where and C is any positively oriented simple closed curve that encloses the origin. F(x, y) = 2хyi + (у? — х?)j (x? + у?)?
> Is the vector field shown in the figure conservative? Explain.
> Match the vector fields F with the plots labeled I–IV. Give reasons for your choices. F (x, y) = 〈x, -y〉 I 3 -3 3 -3
> (a). How do you find the velocity, speed, and acceleration of a particle that moves along a space curve? (b). Write the acceleration in terms of its tangential and normal components.
> 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) = ex sin yz i + zex cos yz j + yex cos yz k
> (a). Write the general form of a second-order homogeneous linear differential equation with constant coefficients. (b). Write the auxiliary equation. (c). How do you use the roots of the auxiliary equation to solve the differential equation? Write the fo
> 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. 1. If y1 and y2 are solutions of y'' + y = 0, then y1 + y2 is also a solution of the equation. 2. I
> Test whether the regression explained by the model in Exercise 12.5 on page 450 is significant at the 0.01 level of significance. 1 Exercise 12.5: The electric power consumed each month by a chemical plant is thought to be related to the average ambient
> Test whether the regression explained by the model in Exercise 12.1 on page 450 is significant at the 0.01 level of significance. Exercise 12.1: A set of experimental runs was made to determine a way of predicting cooking time y at various values of ov
