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.
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The best part? 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? 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. 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. 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. 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. 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. 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.
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.
Vector Analysis Schaum's Outline Book
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, The fact that we cannot visualize points in spaces of dimension higher than three has of course nothing whatsoever to do with their existence. Let x1, x2, Then conversely to each set of coordinates x1, x2, An even shorter notation is simply to write it as ajxi, where we adopt the convention that whenever an index subscript or superscript is repeated in a given term we are to sum over that index from 1 to N unless otherwise specified.
This is called the summation convention. Clearly, instead of using the index j we could have used another letter, say p, and the sum could be written aoxO. Any index which is repeated in a given term, so that the summation convention applies, is called a dummy index or umbral index. An index occurring only once in a given term is called a free index and can stand for any of the numbers 1, 2, If N quantities A1, A2, To provide motivation for this and later transformations, see Problems 33 and 34 of Chapter 7.
If N quantities A1i A2, Note that a superscript is used to indicate contravariant components whereas a subscript is used to indicate covariant components; an exception occurs in the notation for coordinates.
No confusion should arise from this. A scalar or invariant is also called a tensor of rank zero. If to each point of a region in N dimensional space there corresponds a definite tensor, we say that a tensor field has been defined.
This is a vector field or a scalar field according as the tensor is of rank one or zero. It should be noted that a tensor or tensor field is not just the set of its components in one special coordinate system but all the possible sets under any transformation of coordinates.
A tensor is called symmetric with respect to two contravariant or two covariant indices if its components remain unaltered upon interchange of the indices.
If a tensor is symmetric with respect to any two contravariant and any two covariant indices, it is called symmetric. A tensor is called skew-symmetric with respect to two contravariant or two covariant indices if its components change sign upon interchange of the indices. If a tensor is skew-symmetric with respect to any two contravariant and any two covariant indices it is called skew-symmetric.
Vector Analysis Schaum's Outline Book
The sum of two or more tensors of the same rank and type i. Addition of tensors is commutative and associative. The difference of two tensors of the same rank and type is also a tensor of the same rank and type.
Outer Multiplication. The product of two tensors is a tensor whose rank is the sum of the ranks of the given tensors. This product which involves ordinary multiplication of the components of the tensor is called the outer product. However, note that not every tensor can be written as a product of two tensors of lower rank. For this reason division of tensors is not always possible. If one contravariant and one covariant index of a tensor are set equal, the result indicates that a summation over the equal indices is to be taken according to the summation convention.
This resulting sum is a tensor of rank two less than that of the original tensor. The process is called contraction. Inner Multiplication.
By the process of outer multiplication of two tensors followed by a contraction, we obtain a new tensor called an inner product of the given tensors. The process is called inner multiplication. Inner and outer multiplication of tensors is commutative and associative. Quotient Law. Suppose it is not known whether a quantity X is a tensor or not. If an inner prod- uct of X with an arbitrary tensor is itself a tensor, then X is also a tensor.
This is called the quotient law. A matrix of order m by n is an array of quantities apq, called elements, arranged in m rows and n columns and generally denoted by all a The diagonal of a square matrix containing the elements ass, ate, A square matrix whose elements are equal to one in the principal diagonal and zero else h is called a unit matrix and is denoted by 1.
A null matrix, denoted by 0, is a matrix all of whose elements are zero. Matrices whose product is defined r. In general, multiplication of matrices is not commutative, i. AB A BA. Also the distributive laws hold, i. A necessary and sufficient condition that A-1 exist is that det A 0.
The product of a scalar?. The transpose of a matrix A is a matrix AT which is formed from A by interchanging its rows and columns. The transpose of A is also denoted by A.
In rectangular coordinates x,y,z the differential are length ds is obtained from By transforming to general curvilinear coordinates see Problem 17, Chapter 7 this becomes ds 3 3 E I goq dupduq. Such spaces are called three dimensional Euclidean spaces.
In the general case, however, the space is called Riemannian. The quantities gpq are the components of a covariant tensor of rank two called the metric tensor or fundamental tensor. We can and always will choose this tensor to be symmetric see Problem Define g pq byg pq cofactor of gpq gpq g Then gpq is a symmetric contravariant tensor of rank two called the conjugate or reciprocal tensor of gpq see Problem Given a tensor, we can derive other tensors by raising or lowering indices.
For example, given the tensor A pq we obtain by raising the index p, the, tensor A. By raising the index q also we obtain. Where no confusion can arise we shall often omit the dots; thus Apq can be written Apq. These derived tensors can be obtained by forming inner products of the given tensor with the metric tensor g pq or its conjugate gpq. Thus, for example rp p A. Similarly we interpret multiplication by grq as meaning: All tensors obtained from a given tensor by forming inner products with the metric tensor and its conjugate are called associated tensors of the given tensor.
For example A'4 and A. The quantity APBP , which is the inner product of AP and Bq , is a scalar anal- ogous to the scalar product in rectangular coordinates. The symbols are called the Christoffel symbols of the first and second kind respectively. Other symbols used inand 1 q. The latter symbol suggests however a tensor character, which stead o is not trut-tf eneral. As examples, the geodesics on a plane are straight lines whereas the geodesics on a sphere are arcs of great circles.
The covariant derivative of a tensor Ap with respect to x9 is denoted by Apq and is defined by Ap q - 1P1 ,AS aAp axq qs a mixed tensor of rank two. For rectangular systems, the Christoffel symbols are zero and the covariant derivatives are the usual partial derivatives. Covariant derivatives of tensors are also tensors see Problem The above results can be extended to covariant derivatives of higher rank tensors. Thus APi Pin i In performing the differentiations, the tensors g pq , gpq and 80 maybe treated as constants since their covariant derivatives are zero see Problem Since covariant derivatives express rates of change of physical quantities independent of any frames of reference, they are of great importance in expressing physical laws.
Further, let us define It can be shown that E pqr and Epgr are covariant and contravariant tensors respectively, called permutation tensors in three dimensional space. Generalizations to higher dimensions are possible.
The divergence of AP is the contraction of its covariant derivative with respect to xg, i. The curl is also defined as -- Epgr Ap,q. Intrinsic derivatives of higher rank tensors are similarly defined. A tensor Ag The operations of addition, multiplication, etc. See for example Problem Write each of the following using the summation convention.
Write the terms in each of the following indicated sums. Since these indices are associated with the z coordinates and since indices i, j, k are associated respectively with indices p, q, r the required transformation is easily written. A quantity A j, k, 1, m which is a function of coordinates xx transforms to another coordinate system z2 according to the rule axj azk axi a.
Determine whether each of the following quantities is a tensor. If so, state whether it is contravariant or covariant and give its rank: Then is J axi azj axk ax7 axk azk axk axi axk a covariant tensor of rank one or a covariant vector.
We refer to the tensor or equivalently, the tensor with components a- , as the gradient of , written grad 0 or VO.
A covariant tensor has components xy, 2y- z2, xz in rectangular coordinates. Find its covariant components in spherical coordinates. Show that aAp axq is not a tensor even though Ap is a covariant tensor of rank one. By hypothesis, A, -ax, axq aAj axk Ap. Differentiating with respect to -k. Later we axq shall show how the addition of a suitable quantity to aAp ax q causes the result to be a tensor Problem Show that the velocity of a fluid at any point is a contravariant tensor of rank one.
In the coordt dinate system x the velocity is diJ. But d'l axl dxk axk dt dt by the chain rule, and it follows that the velocity is a contravariant tensor of rank one or a contravariant vector. Evaluate a 8q ASS, b bq 8q. Coordinates xP are functions of coordinates xq which are in turn functions of coordinates xr. Then by the chain rule and Problem 11, This az axq az j' indicates that in the transformation equations for the tensor components the quantities with bars and quantities without bars can be interchanged, a result which can be proved in general.
Prove that 8Q is a mixed tensor of the second rank. This is however not a covariant tensor of the second rank as the notation would seem to indicate. If Apq and Brq are tensors, prove that their sum and difference are tensors. Ps Let Ars t be a tensor. What is its rank? Place the corresponding indices j and n equal to each other and sum over this index.
Then We must show that A A' lTnj axJ ark axr axs at pq axp axg azl 'ax--M axg Arst axt are axk axr axs axg ax 1' axg ax l axm vq A rs t 8t axk axr axs Abq 0 axg art arm rst ark axr axs Apq axg ax l axm rsp and so Arse is a tensor of rank 3 and can be denoted by Bqs. The process of placing a contravariant index equal to a covariant index in a tensor and summing is called contraction.Thus APi Prove that if the components of two tensors are equal in one coordinate system they are equal in all coordinate systems.
From a and b we see that the sets of vectors a, b, c and a', b', c' are reciprocal vectors. The curl of the gradient of 0 is zero. Just post a question you need help with, and one of our experts will provide a custom solution. Can I get help with questions outside of textbook solution manuals?
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