Evaluate the integral II (curl F) 0 n dA directly for the given F

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.) F = [y3, x3], C the parabola y = 5x2 from A: (0, 0) to B: (2,20)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.) F as in Prob. 1, C the shortest path from A to B. Is the integral smaller? Give reason.

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F as in Prob. 1, C from A straighL to (2. 0). then vertically up to B

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F = [x2, y2, 0], C the semicircle from (2, 0) to (-2.0), y ~ 0

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.) F = [xy2, x~], C: r = [cosh t, sinh t, 0], o ~ t ~ 2. Sketch C.

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.) F = [ex, eY ] clockwise along the circle with center (0, 0) from (1, 0) to (0, -1)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.) F = [z, x, y], C: r = [cos t, sin t, t] from (1, 0, 0) to (1, 0, 417)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F = [coshx, sinhy. eZ ]. C: r = [t. P, t 3 ] from (0, 0, 0) to (!, i, ~)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F as in Prob. 8. C the straight segment from (0. O. 0) to (!, i, ~)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F = [x, -z, 2y] from (0, 0, 0) straight to (1, 1,0), then to (1, 1, 1), back to (0, 0, 0)

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F = [ex, eY , e Z ], r = [t, P, t 2 ] from (0, 0, 0) to (2, 4, 4). Sketch C.

Calculate f F(r) dr for the following data. If F is a force. c this gives the work done in the displacement along C. (Show the details.)F = [ex, eY , e Z ], r = [t, P, t 2 ] from (0, 0, 0) to (2, 4, 4). Sketch C.

WRITING PROJECT. From Definite Integrals to Line Integrals. Write a short report (1-2 pages) with examples on line integrals as generalizations of definite integrals. The latter give the area under a curve. Explain the corresponding geometric interpretation of a line integral.

PROJECT. Independence of Representation. Dependence on Path. Consider the integral f F(r) dr, where F = [xy, _y2]. C (a) One path, several representations. Find the value of the integral when r = [cos t, sin t], 0 ~ t ~ 1712. Show that the value remains the same if you set t = - p or t = p2 or apply two other parametric transformations of your own choice. (b) Several paths. Evaluate the integral when C: y = x n , thus r = [t, t"l, 0 ~ t ~ 1, where n = 1,2,3, .... Note that these infinitely many paths have the same endpoints. (c) Limit. What is the limit in (b) as n --+ oo? Can you confirm your result by direct integration without referring to (b)? (d) Show path dependence with a simple example of your choice involving two paths.

Evaluate (8) or (8*) with F or f and C as follows.f = x2 + y2, c: r = [t, 4t, 0], 0 ~ t ~ 1

Evaluate (8) or (8*) with F or f and C as follows. f = 1 - sinh2 x, C the catenary r = [t, cosh t], 0~t32

Evaluate (8) or (8*) with F or f and C as follows.F = [y2, Z2, X2], C the helix [3 cos t, 3 sin t, 2t], 0 3 t ~ 817

Evaluate (8) or (8*) with F or f and C as follows.F = [(xy)1/3, (y/x) 1/3, 0], C the hypocycloid r = [cos3 t. sin3 t, 0]. 0 ~ t ~ 17/4

(ML-Inequality, Estimation of Line Integrals) Let F be a vector function defined on a curve C. Let IFI be bounded. say. IFI ~ M on C, where M is some positive number. Show that(9) (L = Length of 0.

Using (9), find a bound for the absolute value of the work W done by the force F = [x2 , y] in the displacement along the segment from (0. Q) to (3, 4).

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show that the fonn under the integral sign is exact in the plane (Probs. 1-4) or in space (Probs. 5-8) and evaluate the integral. (Show the details of your work.)

Show thar in Example 4 of the text we have F = grad (arctan (ylx. Give examples of domains in which the integral is path independent.

PROJECT. Path Dependence. (a) Show that I = 1 (x\ dx + 2xy2 dy) is path dependent in the c xy-plane. (b) Integrate from (0. 0) along the straight-line segment to (1. b). 0 ~ b ~ I, and then vertically up to (I, I); see the figure. For which of these paths is I maximum? What is its maximum value? (c) Integrate from (0, 0) along the straight-line segment to (c, I), 0 ~ c ~ I, and then horizontally to (1, I). For c = I, do you get the same value as for b = I in (b)? For which c is I maximum? What is its maximum value? y 1 (c,l) (1, b) <0.0) 1 x Project 10. Path Dependence

and, if independent, integrare from (0, 0, 0) to (a, b, c).(cosh x::)(:: dx + t' dz)

and, if independent, integrare from (0, 0, 0) to (a, b, c).(3x 2e 2Y + x) dx + 2x3e 2Y dy

and, if independent, integrare from (0, 0, 0) to (a, b, c).3x2 y {lY + x 3 dy + Y d:

and, if independent, integrare from (0, 0, 0) to (a, b, c).2x sin J dx + x 2 cosy dy + y2 dz

and, if independent, integrare from (0, 0, 0) to (a, b, c). (ze X - eY) dx - xeY dy + eX dz

and, if independent, integrare from (0, 0, 0) to (a, b, c).eX cos 2.1' dx - 2ex sin 2y dy - xz dz

and, if independent, integrare from (0, 0, 0) to (a, b, c). xy Z2 d.1. + !X2Z2 dy + x 2)'z do;:

and, if independent, integrare from (0, 0, 0) to (a, b, c). yz cosh x dx + Z sinh x dy + J sinh x dz

and, if independent, integrare from (0, 0, 0) to (a, b, c).Y dt' + (x - 2y) dy + 4x dz

WRITING PROJECT. Ideas on Path Independence. Make a list of the main ideas on path independence and dependence in this section. Then work this list into an essay. including explanations of all definitions and on the practical usefulness of the theorems, but no proofs. Include illustrating examples of your own. Explain what happens in Example 4 if you take the domain 0 < V r + y2 < ~.

(Mean value theorem) lllustrate (2) with an example.

Describe the region of integration and evaluate. (Show the details.)f f (x + y)2 dy dx

Describe the region of integration and evaluate. (Show the details.)10 Ix" (1 - 2xy) dy dx

Describe the region of integration and evaluate. (Show the details.) As Prob. 3, order reversed

Describe the region of integration and evaluate. (Show the details.) 11 cosh tx + y) dx dy

Describe the region of integration and evaluate. (Show the details.)As Prob. 5, order reversed

Describe the region of integration and evaluate. (Show the details.) 1 J e x + 2y dy dx

Describe the region of integration and evaluate. (Show the details.)f f x 2 )' dy dx

Describe the region of integration and evaluate. (Show the details.)I f eX cos y dx dy

Integrate xyeX2 _ y2 over the triangular region with vertices (0. 0). O. I). (1. 2).

Find the volume of the following regions in space.The region beneath z = x2 + y2 and above the square with vertices (1, I), (-I, 1), (-1, -I), (1, -1)

Find the volume of the following regions in space.The tetrahedron cut from the first octant by the plane ~x + 2)' + z = 6. Check by vector methods

Find the volume of the following regions in space.The first octant section cut from the region inside the cylinder r + Z2 = 1 by the planes y = 0, z = 0, x = y

Find the center of gravity (x, y) of a mass of density j(x, y) = I in the given region R.

Find the center of gravity (x, y) of a mass of density j(x, y) = I in the given region R.

Find the center of gravity (x, y) of a mass of density j(x, y) = I in the given region R.

Find the moments of ineltia Ix, Iy , 10 of a mass of density j(x, y) = 1 in the region R shown in the figures (which the engineer is likely to need. along with other profiles listed in engineering handbooks).

Find the moments of ineltia Ix, Iy , 10 of a mass of density j(x, y) = 1 in the region R shown in the figures (which the engineer is likely to need. along with other profiles listed in engineering handbooks).

Find the moments of ineltia Ix, Iy , 10 of a mass of density j(x, y) = 1 in the region R shown in the figures (which the engineer is likely to need. along with other profiles listed in engineering handbooks).

Find the moments of ineltia Ix, Iy , 10 of a mass of density j(x, y) = 1 in the region R shown in the figures (which the engineer is likely to need. along with other profiles listed in engineering handbooks).

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using Green's theorem, evaluate f F(r)-drcounterclockwise c around the boundary curve C of the region R, where

Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In boundary curve C of the region R.

Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In boundary curve C of the region R.

Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In boundary curve C of the region R.

Using (9). evaluate 1, ~w ds counterclockwise over the Jc [In boundary curve C of the region R.

CAS EXPERIMENT. Apply (4) to figures of your choice whose area can also be obtained by another method and compare the results.

(Laplace's equation) Show that for a solution w(x, y) of Laplace's equation \,2U- = 0 in a region R with boundary curve C and outer unit normal vector n, (10) 1, dw = J c w an ds

Show that w = 2ex cos)' satisfies Laplace's equation V2 w = 0 and. using (0), integrate w(ilwldll) counterclockwise around the boundary curve C of the square 0 ~ x ~ 2, 0 ~ y ~ 2.

PROJECT. Other Forms of Green's Theorem in the Plane. Let Rand C be as in Green's theorem, r' a unit tangent vector. and n the outer unit normal vector of C (Fig. 238 in Example 4). Show that (1) may be written or (12) I I(CUrIF)OkdXdy = f For' ds R C (11) I I div F dx dy = f F 0 n ds R C where k is a unit vector perpendicular to the xy-plane. Velify (11) and (12) for F = [7x, - 3)'] and C the circle x 2 + )'2 = 4 as well as for an example of your own choice.

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves II = eonst and u = eonst) of the surface and a nonmal vector N = rll x rv of the surface. (Show the details of your work.)

CAS EXPERIMENT. Graphing Surfaces, Dependence on a, b, c. Graph the surfaces in Probs. 1-10. In Probs. 6--9 generalize the surfaces by introducing parameters a. b. c and then find out in Probs. 3-10 how the shape of the surfaces depends on a. h. c.

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Plane 5x + y - 3z = 30

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.) Plane 4x - 2y + 10.;: = 16

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Sphere (x - 1)2 + (y + 2)2 + Z2 = 25

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Sphere Ix + 2)2 + y2 + (z - 2)2 = I

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Elliptic paraboloid.;: = 4x2 + \.2

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Parabolic cylinder z = 3)'2

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.) Hyperbolic cylinder 9x2 - 4.\'2 = 36

Find a parametric representation and a normal vector. (The answer gives one of them. There are many.)Elliptic cone z = V 9x2 + y2

(Representation z = f(x,y)) Show that z = f(x, y) or g = z - flx, )'J = 0 can be written (f u = ilf/ilu. etc.) (6) r(Lt, v) = [Lt, v, feLt. v)j and N = gradg = I-f,,, -fv, 1].

(Orthogonal parameters) Show that the parameter curves II = const and v = COllst on a surface r(lI, v) are orthogonal (intersect at right angles) if and only if rU.-rv = o.

(Condition (4)) Find the points in Probs. 2-7 at which (4) N * 0 does not hold and state whether this is owing to the shape of the surface or to the choice of the representation.

(Change of representation) Represent the paraboloid in Proh. 4 so that N(O, 0) * 0, and show N.

PROJECT. Tangent Planes T(P) will be less important in our work, bur you should know how to represent them. (a) If S: rILl, V), then T(P): (r* - r ru rv) = 0 (a scalar triple product) or r*(p. q) = rIP) + pr,,(P) + qrvCP). lb) If S: g(x, y, z) = 0, then T(P): (r* - rtp) - v g = O. (c) If S: z = f(x, y), then T(P): z* - z = (x* - x)fx(P) + (y* - y)fY(P)). Interpret (a)-(c) geometrically. Give two examples for (a), two for (b), and two for (c).

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)F = [2x, 5-", 0]. s: r = [u, v, 4u + 3v]. 0~1I~1,-8~v~8

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details of your work.)

CAS EXPERIMENT. Write a program for evaluating sUiface integrals (3) that prints intennediate results (F, F N, the integral over one of the two variables). Can you experimentally obtain rules on functions and surfaces giving integrals that can be evaluated by the usual methods of calculus? Make a list of positive and negative results.

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

Evaluate these integrals for the following data. Indicate the kind of surface. (Show the details.)

(Fun with Mobius) Make Mobius strips from long slim rectangles R of grid paper (graph paper) by pasting the short sides together after giving the paper a halftwist. In each case count the number of parts obtained by cutting along lines parallel to the edge. (a) Make R three squares wide and cut until you reach the beginning. (b) Make R four squares wide. Begin cutting one square away from the edge until you reach the beginning. Then cut the portion that is still two squares wide. (c) Make R five squares wide and cut similarly. (d) Make R six squares wide and cut. Formulate a conjecture about the number of parts obtained.

(Center of gravity) Justity the following formulas for the mass M and the center of gravity (x, y, Z) of a lamina S of density (mass per unit area) u(x, y, z) in space: M= II udA. x= ~ If.rudA, s s :v = ~ I I yudA, Z = ~ II zudA.

(Moments of inertia) Justify the following formulas for the moments of inertia of the lamina in Prob. 22 about the x-, y-. and ::;-axes. respectively: s s Iz = II (x2 + y2)udA.

Find a fonnula for the moment of inertia of the lamina in Prob. 22 about the line y = x, ::; = O. Find the moment of inertia of a lamina S of density 1 about an axis A, where

S: x2 + y2 = I, 0 ~ z ~ h, A: the z-axis

S as in Prob. 25. A: the line::: = h/2 in the xc-plane

S: x2 + y2 = Z2, 0 ~ Z ~ h, A: the z-axis

(Steiner's theorem6) If IA is the moment of inertia of a mass distribution of total mass M with respect to an axis A through the center of gravity, show that its moment of inertia IB with respect to an axis E, which is parallel to A and has the distance k from it. is

Using Steiner's theorem, find the moment of inertia of S in Prob. 26 about the x-axis.

TEAM PROJECT. First Fundamental Form of a Surface. Given a surface S: r(lI, v), the corresponding quadratic differential fonn (13) ds2 = E du2 + 2F du dv + G dv2 with coefficients is called the first fundamental form of S. (E, F, G are standard notations that have nothing to do with F and G that occur at some other places in this chapter.) The first fundamental form is basic in the theory of surfaces, since with its help we can determine lengths, angles, and areas on S. To show this, prove the following. (a) For a curve C: u = u(t), v = vet), a ~ t ~ b, on S, formulas (10), Sec. 9.5, and (14) give the length (15) b I = I v'r'(t).r'(t) dt a b = I YEu'2 + 2Fu'v' + Gv'2dt. a (b) The angle 'Y between two intersecting curves Cr: u = gUY, v = h(t) and C2 : u = p(t), v = q(t) on S: r(u, v) is obtained from (16) cos 'Y = where a = rug' + rvh' and b = rup' + rvq' are tangent vectors of Cr and C2 . (c) The square of the length of the normal vector N can be written so that formula (8) for the area A(S) of S becomes A(S) = I I dA = I I INI du dv (18) S R = I I Y EG - F2 du dv. R (d) For polar coordinates u (= r) and v (= 8) defined by x = u cos v, y = u sin v we have E = 1, F = O. G = u2 , so that ds 2 = du2 + u2 dv2 = dr2 + r2 d82. Calculate from this and (18) the area of a disk of radius a. (e) Find the tirst fundamental fOlm of the torus in Example 5. Use it to calculate the area A of the torus. Show that A call also be obtained by the theorem of pos,7 which states that the area of a surface of revolution equals the product of the length of a meridian C and the length of the path of the center of gravity of C when C is rotated through the angle In. (I) Calculate the first fundamental form for the usual representations of important surfaces of your own choice (cylinder, cone, etc.) and apply them to the calculation of length~ and areas on these ~urfaces.

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.)u = x2y2~2, T the box Ixl ~ a, iYI ~ b, kl ~ c

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.) u = x2 + y2 + ~2, T the box 0 ~ x ~ 4, 0 ~ J ~ 9, O~:::~l

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.) u = sin x cos y, T: 0 ~ x ~ ~7T. ~7T - X ~ Y ~ ~7T, o ~ ~ ~ 12

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.) u = e-"'-Y-z, Tthe tetrahedron with vertices (0. O. 0). (2, O. a), (0, 2. a). (0, 0, 2)

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.)u = ~(X2 + y2)2, T the cylinder x2 + )'2 ~ 4. Izl ~ 2

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.)u = 30::;. T the region in the first octant bounded by y = 1 - x 2 and z = x. Sketch it.

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.). u = 1 + Y + ~2, T the cylinder ,,2 + ::.2 ~ 9, l~x~9

Find the total mass of a mass distribution of density u in a region T in space. (Show the details of your work.)u = x 2 + y2. T the ball x 2 + y2 + :;2 ~ a2

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Ix = J J J (y2 + z2) dx dy dz of a mass of density 1 in T a region T about the x-axis. Find Ix when T is as follows.

Show that for a solid of revolution, f" = 27T L r\x) dx. U~t: this to solve Probs. 11-14.

Why is Tx in Prob. 13 for large h larger than I,,, in Prob. 14? Why is it smaller for h = I? Give physical reason.

Evaluate this integral by the divergence theorem. (Show the details.)F = [x, y, ;::], S the sphere x 2 + y2 + ::;2 = 9

Evaluate this integral by the divergence theorem. (Show the details.)F = [4x, 3z, 5y]. S the surface of the cone x 2 + y2 ~ :;2. 0 ~ ~ ~ 2

Evaluate this integral by the divergence theorem. (Show the details.)F = [z - y y3, 2;::3]. S the surface of y2 + Z2 ~ 4, -3 ~x~

Evaluate this integral by the divergence theorem. (Show the details.) F = [3x)' 2, yx2 - y3, 3zx2], S the surface of t 2 + y2 ~ 25. 0 ~ Z ~ 2

Evaluate this integral by the divergence theorem. (Show the details.)F = [sin y, cos x. cos ;::], S tile surface of x 2 + )'2 ~ 4. Izl ~ 2

Evaluate this integral by the divergence theorem. (Show the details.)F = [x3 - .\'3, )'3 - Z3, ;::3 - x3 ]. S the surface of x 2 + y2 + ;::2 ~ 25, z :0=: 0

Evaluate this integral by the divergence theorem. (Show the details.)F = [4x2 2x + y2. x 2 + Z2]. S the surface of the tetrahedron in Prob. 4

Evaluate this integral by the divergence theorem. (Show the details.)F = [4x2 y2. -2 cos m:], S the surface of the tetrahedron with vertices (0, O. a). (l. O. a). (0. l. m. (0,0, 1)

Evaluate this integral by the divergence theorem. (Show the details.)F = [5x 3. 5y 3, 5;::3], S: x 2 + y2 + Z2 = 4

(Hannonic functions) Verify Theorem 1 for f = 2x2 + 2y2 - 4z2 and S the surface of the cube o ~ x ~ I, 0 ~ y ~ 1, 0 ~ z ~ 1

(Hannonic functions) Verify Theorem 1 for f = y2 - x 2 and the surface of the cylinder x 2 + y2 ~ I, 0 ~ z ~ 5.

(Green's first formula) Verify (8) for f = 3y2, g = x 2 , S the surface of the cube in Prob. I.

(Green's first formula) Verify (8) for f = x, g = y2 + ;:2. S the surface of the box 0 ~ x ~ 1, o ~ Y ~ 2, 0 ~ z ~ 3.

(Green's second formula) Verify (9) for the data in Prob.3.

(Green's second formula) Verify (9) for f = x4, g = y2 and the cube in Prob. l.

(Volume as a surface integral) Show that a region T with boundary surface S has the volume V= ~ IJrcostPdA 3 S where r is the distance of a variable point P: (x, y, z) on S from the origin 0 and tP is the angle between the directed line OP and the outer normal of Sat P.(Make a sketch.)

Find the volume of a ball of radius a by means of the formula in Prob. 7.

Show that a region T with boundary surface S has the volume V= IIXdydz S = IIyd::dx

TEAM PROJECT. Divergence Theorem and Potential Theory. The importance of the divergence theorem in potential theory is obvious from (7)-(9) and Theorems I - 3. To emphasize it further, consider functions f and g that are harmonic in some domain D containing a region Twith boundary surface S such that T satisfies the assumptions in the divergence theorem. Prove and illustrate by examples that then: (a) II g :! dA = I IIlgrad gl2 dV. S T (b) If aglan = 0 on S, then g i8 constant in T. (c) II ( f og - g Of) dA = O. on 011 S (d) If of Ian = fJglon on S, then f = g + c in T, where c is a constant. (e) The Laplacian can be represented independently of coordinate systems in the form v2 = lim _1_ JI of dA f d(T)~O VeT) an S(T) where d(T) is the maximum distance of the points of a region T bounded by SeT) from the point at which the Laplacian is evaluated and veT) is the volume of T.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Evaluate the integral II (curl F) 0 n dA directly for the given F and S.

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

Calculate this line integral by Stokes's theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be righthanded. (Show the details.)

(Stokes's theorem not applicable) Evaluate f Fo r' ds, c F = (x2 + y2)-1[ -y,x], C: x 2 + y2 = I, z = 0, oriented clockwise. Why can Stokes's theorem not be applied? What (false) result would it give?

WRITING PROJECT. Grad, Div, Curl in Connection with Integrals. Make a list of ideas and results on this topic in this chapter. See whether you can rearrange or combine parts of your material. Then subdivide the material into 3-5 portions and work out the details of each portion. Include no proofs but simple typical examples of your own that lead to a better understanding of the material.

List the kinds of integrals in this chapter and how the integral theorems relate some of them.

How can work of a variable force be expressed by an integral?

State from memory how you can evaluate a line integral. A double integral.

What do you remember about path independence? Why is it important?

How did we Use Stokes's theorem in connection with path independence?

State the definition of curl. Why is it important in this chapter?

How can you transform a double integral or a surface integral into a line integral?

What is orientation of a surface? What is its role in connection with surface integrals?

State the divergence theorem and its applications from memory.

State Laplace's equation. Where in physics is it important? What properties of its solutions did we discuss?

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Evaluate. with F and C as given, by the method that seems most suitable. Recall that if F is a force, the integral gives the work done in a displacement along C. (Show the details.)

Find the coordinmes .i. y of the center of gravity of a mass of density I(x. y) in the region R. (Sketch R. Shmv the details.)

Find the coordinmes .i. y of the center of gravity of a mass of density I(x. y) in the region R. (Sketch R. Shmv the details.)

Find the coordinmes .i. y of the center of gravity of a mass of density I(x. y) in the region R. (Sketch R. Shmv the details.)

Find the coordinmes .i. y of the center of gravity of a mass of density I(x. y) in the region R. (Sketch R. Shmv the details.)

Find the coordinmes .i. y of the center of gravity of a mass of density I(x. y) in the region R. (Sketch R. Shmv the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)

Evaluate this integral directly or. if pos~ible. by the divergence theorem. (Show the details.)