# VECTOR ANALYSIS SCHAUM SERIES SOLUTION PDF

pdf. Vector analysis Schaum series/ schaum's outline. Pages Show that the general solution of the differential equation d stants, is (a) r = e-at (C 1 a. Vector Analysis Schaum's Outline Book - Free ebook download as PDF File Vector Analysis, Schaum's outlines, fully solved problems. Solutions. Get instant access to our step-by-step Schaum's Outline Of Vector Analysis, 2ed downloaded Schaum's Outline of Vector Analysis, 2ed PDF solution manuals?.

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Get This Link to read/download book >>> Vector Analysis, 2nd Edition More than a vector analysis solution of Schaum's Outline Series for free as a PDF file?. [PDF] Schaum's Outlines Vector Analysis (And An Introduction to Tensor Analysis ) 1st Edition Confusing Textbooks? Missed Lectures? Not Enough Time?. Sep 5, Save this Book to Read schaum outlines vector analysis solution manual PDF eBook at our Online Library. Get schaum outlines vector analysis.

If 0 x,y,z is a scalar invariant with respect to a rotation of axes, prove that grad q5 is a vector invariant under this transformation. Evaluate V 3r If U is a differentiable function of x,y,z , prove W.

Find the directional derivative of Ans. This determinant is called the Jacobian of u,v,w with reu, v, w spect to x,y,z and is written or J u,v,w. Evaluate V ln r. Prove Ans. Evaluate V r3 r. Evaluate Find the most general differentiable function f r so that f r r is solenoidal.

Plot and give a physical interpretation. Prove curl 0 grad j. Compute the divergence and curl of each vector field and explain the physical significance of the results obtained. A A V-B. Find cp such that A Ans. If f r is aifferentiable, prove that f r r is irrotational. If so, find V.

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If A x,y,z is an invariant differentiable vector field with respect to a rotation of axes, prove that a div A and b curl A are invariant scalar and vector fields respectively under the transformation.

Solve equations 3 of Solved Problem 38 for x,y,z in terms of x', y', z'. Show that under a rotation 0 i. Show that the Laplacian operator is invariant under a rotation.

We assume that C is composed of a finite number of curves for each of which r u has a continuous derivative. If A is the force F on a particle moving along C, this line integral represents the work done by the force. If C is a closed curve which we shall suppose is a simple closed curve, i. In general, any integral which is to be evaluated along a curve is called a line integral.

Such integrals can be defined in terms of limits of sums as are the integrals of elementary calculus. For methods of evaluation of line integrals, see the Solved Problems. The following theorem is important. C In such case A is called a conservative vector field and cb is its scalar potential. In such case A. See Problems Let S be a two-sided surface, such as shown in the figure below. Let one side of S be considered arbitrarily as the positive side if S is a closed surface this is taken as the outer side.

A unit normal n to any point of the positive side of S is called a positive or outward drawn unit normal. Associate with the differential of surface area dS a vector dS whose magnitude is dS and whose direction is that of n.

The integral ffA. Such integrals can be defined in terms of limits of sums as in elementary calculus see Problem The notation 9j. Where S no confusion can arise the notation may also be used. S To evaluate surface integrals, it is convenient to express them as double integrals taken over the projected area of the surface S on one of the coordinate planes.

This is possible if any line perpendicular to the coordinate plane chosen meets the surface in no more than one point. However, this does not pose any real problem since we can generally subdivide S into surfaces which do satisfy this restriction. Consider a closed surface in space enclosing a volume V. Then ff5.

For evaluation of such integrals, see the Solved Problems. The acceleration of a particle at any time t? Compare with Problem 3. A force directed toward or away from a fixed point 0 and having magnitude depending only on the distance r from 0 is called a central force.

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The area swept out by the position vector in this time is approximately half the area of a parallelogram with sides r and A r, or 2 r x A r. Let m and M be the masses of the planet and sun respectively and choose a set of coordinate axes with the origin 0 at the sun.

According to part c , a planet moves around the sun so that its position vector sweeps out equal areas in equal times. This result and that of Problem 5 are two of Kepler's famous three laws which he deduced empirically from volumes of data compiled by the astronomer Tycho Brahe. These laws enabled Newton to formulate his universal law of gravitation. For Kepler's third law see Problem Show that the path of a planet around the sun is an ellipse with the sun at one focus.

Since r. The orbit is an ellipse, parabola or hyperbola accord- ing as E is less than, equal to or greater than one. Since orbits of planets are closed curves it follows that they must be ellipses. We call this the positive direction, or say that C has been traversed in the positive sense.

If C were traversed in the clockwise negative direction the value of the integral would be - 18 IT. This is true of course only if q5 x,y,z is single-valued at all points P1 and P2. By hypothesis, fc F. F Proof using vectors. F is irrotational. F is irrotational , prove that F is conservative. Let us choose as a particular path the straight line segments from x1, y1, z1 to x,yl, z1 to x,y,z1 to x,y,z and call 0 x,y,z the work done along this particular path.

Prove that if J F. P1 BP2 Suppose a particle of constant mass m to move in this field. The result states that the total energy at A equals the total energy at B conservation of energy. Give a definition of A- n dS over a surface S in terms of limit of a sum.

Choose any point Pp within A So whose coordinates are xp, yp, z,. Let no be the positive unit normal to AS at P. Now take the limit of this sum as M--a in such a way that the largest dimension of each A Sp approaches zero. This limit, if it exists, is called the surface integral of the normal component of A over S and is denoted by ff A n dS S Suppose that the surface S has projection R on the xy plane see figure of Prob.

Show that ffA. The surface S and its projection R on the xy plane are shown in the figure below. In this manner R is completely covered. Project S on the xz plane as in the figure below and call the projection R. Note that the projection of S on the xy plane cannot be used here. Then fJr VxF. Then ff F. Then ffF. In dealing with surface integrals we have restricted ourselves to surfaces which are two-sided. Give an example of a surface which is not two-sided.

Take a strip of paper such as ABCD as shown in the adjoining figure. Twist the strip so that points A and B fall on D and C respectively, as in the adjoining figure. If n is the positive normal at point P of the surface, we find that as n moves around the surface it reverses its original direction when it reaches P again.

If we tried to color only one side of the surface we would find the whole thing colored. This surface, called a Moebius strip, is an example of a one-sided surface. This is sometimes called a non-orientable surface. A two-sided surface is orientable. Define p xk, yk, zk qk.

The limit of this sum, when M--c in such a manner that the largest of the quantities Auk will approach zero, if it exists, is denoted by fff 0 dV. It can be shown that this limit V is independent of the method of subdivision if is continuous throughout V. In forming the sum 1 over all possible cubes in the region, it is advisable to proceed in an orderly fashion. One possibility is to add first all terms in 1 corresponding to volume elements contained in a column such as PQ in the above figure.

This amounts to keeping xk and yk fixed and adding over all zk's. Next, keep xkfixed but sum over all yk's. This amounts to adding all columns such as PQ contained in a slab RS, and consequently amounts to summing over all cubes contained in such a slab. Finally, vary xk.

This amounts to addition of all slabs such as RS. In the process outlined the summation is taken first over zk's then over yk's and finally over xk's. However, the summation can clearly be taken in any other order. Next keep x constant and integrate with respect to y. J1 Ax B dt. Evaluate a Ans. This describes the motion of a projectile fired from a cannon inclined at angle 60 with the positive x-axis with initial velocity of magnitude vo. Prove that the squares of the periods of planets in their motion around the sun are proportional to the cubes of the major axes of their elliptical paths Kepler's third law.

Evaluate f F. T ds where s is the arc length. F dr around the triangle C of Figure 1, a in the indicated Evaluate fr' A dr around the closed curve C of Fig. Show that the work done on a particle in moving it from A to B equals its change in kinetic energies at these points whether the force field is conservative or not. If so, find it. Evaluate ff A. The direction of C is called positive if an observer, walking on the boundary of S in this direction, with his head pointing in the direction of the positive normal to S, has the surface on his left.

Unless otherwise stated we shall always assume f to mean that the integral is described in the positive sense. Green's theorem in the plane is a special case of Stokes' theorem see Problem 4. Also, it is of interest to notice that Gauss' divergence theorem is a generalization of Green's theorem in the plane where the plane region R and its closed boundary curve C are replaced by a space region V and its closed boundary surface S.

For this reason the divergence theorem is often called Green's theorem in space see Problem 4. Green's theorem in the plane also holds for regions bounded by a finite number of simple closed curves which do not intersect see Problems 10 and See Problem Then 5. See Problems 22, 23, and The result proves useful in extending the concepts of gradient, divergence and curl to coordinate systems other than rectangular see Problems 19, 24 and also Chapter 7.

Prove Green's theorem in the plane if C is a closed curve which has the property that any straight line parallel to the coordinate axes cuts C in at most two f points. Verify Green's theorem in the plane for shown in the adjacent diagram.

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Extend the proof of Green's theorem in the plane given in Problem 1 to the curves C for which lines parallel to the coordinate axes may cut C in more than two points. By constructing line ST the region is divided into two regions R.

A region which is not simply-connected is called multiply-connected. We have shown here that Green's theorem in the plane applies to simply-connected regions bounded by closed curves. In Problem 10 the theorem is extended to multiply-connected regions. For more complicated simply-connected regions it may be necessary to construct more lines, such as ST, to establish the theorem. Express Green's theorem in the plane in vector notation.

A generalization of this to surfaces S in space having a curve C as boundary leads quite naturally to Stokes' theorem which is proved in Problem Interpret physically the first result of Problem 4.

If A denotes the force field acting on a particle, then fe A dr is the work done in moving the particle around a closed path C and is determined by the value of Vx A. This amounts to saying that the work done in moving the particle from one point in the plane to another is independent of the path in the plane joining the points or that the force field is conservative.

These results have already been demonstrated for force fields and curves in space see Chapter 5. Conversely, if the integral is independent of the path joining any two points of a region, i. A direct evaluation is difficult. Then we can use any path, for example the path consisting of straight line segments from 0,0 to 2,0 and then from 2,0 to 2,1.

Then 2,1 2,1 10x4 -2xy3 dx - 3x2y2 dy 0, 0 7. Evaluate C triangle of the adjoining figure: Note that although there exist lines parallel to the coordinate axes coincident with the coordinate axes in this case which meet C in an infinite number of points, Green's theorem in the plane still holds.

In general the theorem is valid when C is composed of a finite number of straight line segments. Show that Green's theorem in the plane is also valid for a multiply-connected region R such as shown in the figure below. The boundary of R, which consists of the exterior boundary AHJKLA and the interior boundary DEFGD, is to be traversed in the positive direction, so that a person traveling in this direction always has the region on his left.

It is seen that the positive directions are those indicated in the adjoining figure. In order to establish the theorem, construct a line, such as AD, called a cross-cut, connecting the exterior and interior boundaries. For a generalization to space curves, see Problem Demonstrate the divergence theorem physically.

From Figure a below: Prove the divergence theorem.

## Vector Analysis Schaum's Outline Book

Let S be a closed surface which is such that any line parallel to the coordinate axes cuts S in at most two points. Denote the projection of the surface on the xy plane by R.

Adv or S V The theorem can be extended to surfaces which are such that lines parallel to the coordinate axes meet them in more than two points. To establish this extension, subdivide the region bounded by S into subregions whose surfaces do satisfy this condition. The procedure is analogous to that used in Green's theorem for the plane. Evaluate ff F. By the divergence theorem, the required integral is equal to fffv.

Note that evaluation of the surface integral over S3 could also have been done by projection of S3 on the xz or yz coordinate planes. In Chapter 7 we use this definition to extend the concept of divergence of a vector to coordinate systems other than rectangular.

If div A is positive in the neighborhood of a point P it means that the outflow from P is positive and we call P a source. Similarly, if div A is negative in the neighborhood of P the outflow is really an inflow and P is called a sink. Evaluate jfr. In the proof we have assumed that 0 and scalar functions of position with continuous derivatives of the second order at least. Then fffv. AS Using the same principle employed in Problem 19, we have ffqn. I nxAdS. Multiplying by i, j, k and adding, the result follows.

The results obtained can be taken as starting points for definition of gradient and curl. Using these definitions, extensions can be made to coordinate systems other than rectangular. To establish the equivalence, the results of the operation on a vector or scalar field must be consistent with already established results.

Also if o is ordinary multiplication, then for a scalar 0, Vo lim ffdsoq AS established in Problem 24 a. Let S be a closed surface and let r denote the position vector of any point x,y,z measured from an origin 0.

## Schaum's Outline of Vector Analysis, 2ed Solutions Manual

Prove that ffn. This result is known as Gauss' theorem. S b If 0 is inside S, surround 0 by a small sphere s of radius a. Let 'r denote the region bounded by S and s. Interpret Gauss' theorem Problem 26 geometrically. Let dS denote an element of surface area and connect all points on the boundary of dS to 0 see adjoining figure , thereby forming a cone.

Let S be a surface, as in Figure a below, such that any line meets S in not more than two points. An integration over these two regions gives zero, since the contributions to the solid angle cancel out. S In case 0 is inside S. The total solid angle in this case is equal to the area of a unit sphere which is 47T, so that ffn. If 0 is outside S, for example, then a cone with vertex at 0 intersects S at an even number of places and the contribution to the surface integral is zero since the solid angles subtended at 0 cancel out in pairs.

If 0 is inside S, however, a cone having vertex at 0 intersects S at an odd number of places and since cancellation occurs only for an even number of these, there will always be a contribution of 47T for the entire surface S. A fluid of density p x,y,z,t moves with velocity v x,y,z,t. Then 5ff apat dv V or fffv. If p is a constant, the fluid is incompressible and V.

The total flux of heat across S, or the quantity of heat leaving S per unit time, is ffKvu. For steady-state heat flow i.

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We must show that ff vxA. By Green's theorem for the plane the last integral equals F dx where F is the boundary of R.

For assume that S can be subdivided into surfaces S1,S2, Sk with boundaries C1, C2, Ck which do satisfy the restrictions. Then Stokes' theorem holds for each such surface. Adding these surface integrals, the total surface integral over S is obtained.

Adding the corresponding line integrals over C1, C2, Ck , the line integral over C is obtained. The boundary C of S is a circle in the xy plane of radius one and center at the origin. Prove that a necessary and sufficient condition that A A. Then by Stokes' theorem f C Necessity.

Suppose f ff VXA. Let S be a surface contained in this region whose normal n at each point has the same direction as Ox A , i. Let C be the boundary of S.

See Problems 10 and 11, Chapter 5. This can be used as a starting point for defining curl A see Problem 36 and is useful in obtaining curl A in coordinate systems other than rectangular. Since , A A. If curl A is defined according to the limiting process of Problem 35, find the z component of curl A. Let Al and A2 be the components of A at P in the positive x and y directions respectively. Adding, we have approximately 5 A.

Evaluate f TT. Interpret What restrictions should you make? Illustrate the result where u and v are polar coordinates. Show that Green's second identity can be written fff c15V2qi V A vector B is always normal to a given closed surface S. Show that region bounded by S. If C 2 Ans. Use the operator equivalence of Solved Problem 25 to arrive at a V0, b V.

A, c V x A in rectangular coordinates. Adv Let r be the position vector of any point relative to an origin 0. Suppose 0 has continuous derivatives of order two, at least, and let S be a closed surface bounding a volume V.

## [PDF] Schaumâ€™s Outline of Vector Analysis By Murray R Spiegel, Seymour Lipschutz Book Free Download

Denote 0 at 0 by 0o. The potential O P at a point P x,y,z due to a system of charges or masses gl,g2, Deduce the following under suitable assumptions: Given a point P with rectangular coordinates x, y, z we can, from 2 associate a unique set of coordinates u1, u2, u3 called the curvilinear coordinates of P.

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The sets of equations 1 or 2 define a transformation of coordinates. If the coordinate surfaces intersect at right angles the curvilinear coordinate system is called orthogonal. The u1, u2 and u3 coordinate curves of a curvilinear system are analogous to the x, y and z coordinate axes of a rectangular system. Thus at each point P of a curvilinear system there exist, in general, two sets of unit vectors, e1, e2, e3 tangent to the coordinate curves and E1, E2, E3 normal to the coordinate surfaces see Fig.

The sets become identical if and only if the curvilinear coordinate system is orthogonal see Problem Both sets are analogous to the i, j, k unit vectors in rectangular coordinates but are unlike them in that they may change directions from point to point. It can be shown see Problem 15 that the sets au, au and Vu1, Vu2, Vu3 constitute reciprocal systems vectors.

We can also represent A in terms of the base vectors -6 r Vu1, Vu2, Vu3 which au l. Then the differential of arc length ds1 along u1 at P is h1 du1. Referring to Fig. Extensions of the above results are achieved by a more general theory of curvilinear systems using the methods of tensor analysis which is considered in Chapter 8.

Cylindrical Coordinates p, 0, z. Spherical Coordinates r, 6, 0. Parabolic Cylindrical Coordinates u, v, z. They are confocal parabolas with a common axis. Paraboloidal Coordinates u, v, 0. The third set of coordinate surfaces are planes passing through this axis. Elliptic Cylindrical Coordinates u, v, z. They are confocal ellipses and hyperbolas. Prolate Spheroidal Coordinates ,77, 0. Oblate Spheroidal Coordinates 6,77, qb.

Bipolar Coordinates u, v, z. By revolving the curves of Fig. Describe the coordinate surfaces and coordinate curves for a cylindrical and b spherical coordinates.

The coordinate curves are: Determine the transformation from cylindrical to rectangular coordinates. Such points are called singular points of the transformation.

Prove that a cylindrical coordinate system is orthogonal. Thus determine AO, 4 and Az. Express the velocity v and acceleration a of a particle in cylindrical coordinates. Find the square of the element of arc length in cylindrical coordinates and determine the corresponding scale factors. First Method. Second Method. Work Problem 7 for a spherical and b parabolic cylindrical coordinates. Sketch a volume element in a cylindrical and b spherical coordinates giving the magnitudes of its edges.

Find the volume element dV in a cylindrical, b spherical and c parabolic cylindrical coordinates. Find a the scale factors and b the volume element dV in oblate spheroidal coordinates. Find expressions for the elements of area in orthogonal curvilinear coordinates. Referring to Figure 3, p. We shall therefore require the Jacobian to be different from zero.

To cover the required region in the first octant, fix 8 and 0 see Fig. Here we have performed the integration in the order r, 8, o although any order can be used. In general, when transforming multiple integrals from rectangular to orthogonal curvilinear coordinates the volume element dx dy dz is replaced by h1h2h3 duldu.

If u1, u2, u3 are general coordinates, show that '3r au1 -ar 'a r a u2 ' auand Vu,, Vu2,Vu3 are recipro3 cal systems of vectors. Then the required result follows from Problem 53 c of Chapter 2.

The quantities g0 are called metric coefficients and are symmetric, 2i. The metric form extended to higher dimensional space is of fundamental importance in the theory of relativity see Chapter 8.

Derive an expresssion for v4 in orthogonal curvilinear coordinates. Let u1, u2, u3 be orthogonal coordinates. Show that in orthogonal coordinates V a Al e1 h1 h2h3 au.

Express V2q in orthogonal curvilinear coordinates. In this case the calculation would proceed in a manner analogous to that of Problem 21, Chapter 4. Let us first calculate curl A e1. To do this consider the surface S1 normal to e1 at P, as shown in the adjoining figure. Denote the boundary of S1 by C1. We have PQ fA. Write Laplace's equation in parabolic cylindrical coordinates. Let A be a given vector defined with respect to two general curvilinear coordinate systems ui, u2, u3 and ui, u2, u3.

Find the relation between the contravariant components of the vector in the two coordinate systems. Suppose the transformation equations from a rectangular x, y, z system to the ui, u2i u3 and ui , u2 , i. If three quantities C1, C2, C3 of a coordinate system u1, u2, u3 are related to three other quantities C1, C2, C3 of another coordinate system Z1,2, u3 by the transformation equations 6 , 7 , 8 or 9 , then the quantities are called components of a contravariant vector or a contravariant tensor of the first rank.

Work Problem 33 for the covariant components of A. Write the covariant components of A in the systems u1, u2, u3 and c1, c2, c3 respectively. If three quantities c1, c2, c3 of a coordinate system u1, u2, u3 are related to three other quantities c1 , c2 , c3 of another coordinate system u1, u2, u3 by the transformation equations 6 , 7 , 8 or 9 , then the quantities are called components of a covariant vector or a covariant tensor of the first rank.

In generalizing the concepts in this Problem and in Problem 33 to higher dimensional spaces, and in generalizing the concept of vector, we are led to tensor analysis which we treat in Chapter 8.

In the process of generalization it is convenient to use a concise notation in order to express fundamental ideas in compact form. It should be remembered, however, that despite the notation used, the basic ideas treated in Chapter 8 are intimately connected with those treated in this chapter.

Describe and sketch the coordinate surfaces and coordinate curves for a elliptic cylindrical, b bipolar, and c parabolic cylindrical coordinates. Determine the transformation from a spherical to rectangular coordinates, b spherical to cylindrical coordinates. Express each of the following loci in spherical coordinates: If p, 0, z are cylindrical coordinates, describe each of the following loci and write the equation of each locus in rectangular coordinates: If u, v, z are parabolic cylindrical coordinates, graph the curves or regions described by each of the fol- lowing: Prove that a spherical coordinate system is orthogonal.

Prove that a parabolic cylindrical, b elliptic cylindrical, and c oblate spheroidal coordinate systems are orthogonal. Express the velocity v and acceleration a of a particle in spherical coordinates. Find the square of the element of are length and the corresponding scale factors in a paraboloidal, b elliptic cylindrical, and c oblate spheroidal coordinates.

Find the volume element dV in a paraboloidal, b elliptic cylindrical, and c bipolar coordinates. Find a the scale factors and b the volume element dV for prolate spheroidal coordinates. Derive expressions for the scale factors in a ellipsoidal and b bipolar coordinates. Find the elements of area of a volume element in a cylindrical, b spherical, and c paraboloidal coordinates.

Find the Jacobian J x'y'z u1, u2. Evaluate V Hint: Use cylindrical coordinates. Use spherical coordinates to find the volume of the smaller of the two regions bounded by a sphere of radius a and a plane intersecting the sphere at a distance h from its center.

Find au, t r au2 ,our3 , Du1, Out, Qua in a cylindrical, b spherical, and c parabolic cylindrical co- ordinates. Find TD, div A and curl A in parabolic cylindrical coordinates. Express a V Ji and b V A in spherical coordinates. Find Vq in oblate spheroidal coordinates. Express Maxwell's equation V x E in elliptic cylindrical coordinates. Write Laplace's equation in paraboloidal coordinates.

Find the element of are length on a sphere of radius a. Use R this to determine the surface area of a sphere. Let x, y be coordinates of a point P in a rectangular xy plane and u, v the coordinates of a point Q in a rectangular uv plane.

The result is important in the theory of Laplace transforms. Let x, y, z and u1, u2i u3 be respectively the rectangular and curvilinear coordinates of a point.

The coordinate curves are the intersections of the coordinate surfaces. A study of the consequences of this re-L quirement leads to tensor analysis, of great use in general relativity theory, differential geometry, mechanics, elasticity, hydrodynamics, electromagnetic theory and numerous other fields of science and engineering.

In three dimensional space a point is a set of three numbers, called coordinates, determined by specifying a particular coordinate system or frame of reference. For example x,y, z , p, c,z , r, 8, 55 are coordinates of a point in rectangular, cylindrical and spherical coordinate systems respectively.

A point in N dimensional space is, by analogy, a set of N numbers denoted by x1, x2, As a Chegg Study subscriber, you can view available interactive solutions manuals for each of your classes for one low monthly price. Why download extra books when you can get all the homework help you need in one place? Can I get help with questions outside of textbook solution manuals? You bet! Just post a question you need help with, and one of our experts will provide a custom solution.

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Unlike static PDF Schaum's Outline of Vector Analysis, 2ed solution manuals or printed answer keys, our experts show you how to solve each problem step-by-step. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. You can check your reasoning as you tackle a problem using our interactive solutions viewer. Plus, we regularly update and improve textbook solutions based on student ratings and feedback, so you can be sure you're getting the latest information available.

Our interactive player makes it easy to find solutions to Schaum's Outline of Vector Analysis, 2ed problems you're working on - just go to the chapter for your book. Hit a particularly tricky question? Bookmark it to easily review again before an exam. The best part?The results obtained can be taken as starting points for definition of gradient and curl.

Interpret Gauss' theorem Problem 26 geometrically. Compare with Problem Oblate spheroidal coordinates. By constructing line ST the region is divided into two regions R. Determine the Christoffel symbols of the second kind in a rectangular, b cylindrical, and c spherical coordinates.

The quantities gpq are the components of a covariant tensor of rank two called the metric tensor or fundamental tensor. The term is used by others in a slightly weaker sense. If so, find it. On the other hand any three or fewer of these vectors are linearly independent.

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