Vector Analysis and Tensor Analysis: An Introduction with Schaum's Exercises and Solutions

Tensor Analysis Schaum Series Pdf 38

Tensor analysis is a branch of mathematics that deals with quantities that have magnitude and direction in multiple dimensions. It is widely used in physics and engineering to describe phenomena such as forces, stresses, strains, electric and magnetic fields, fluid dynamics, relativity and quantum mechanics. In this article, we will introduce the basic concepts and applications of tensor analysis, and show you how to download the pdf file of a popular book on this topic: Schaum's outline of theory and problems of vector analysis and an introduction to tensor analysis by Seymour Lipschutz, Dennis Spellman and Murray R. Spiegel.

Tensor Analysis Schaum Series Pdf 38

Introduction

What is tensor analysis?

A tensor is a generalization of a vector or a matrix that can represent any physical quantity that depends on multiple coordinates or directions. For example, a scalar is a tensor of rank zero that has only magnitude, such as temperature or mass. A vector is a tensor of rank one that has magnitude and direction in one dimension, such as velocity or force. A matrix is a tensor of rank two that has magnitude and direction in two dimensions, such as stress or strain. A tensor of rank three has magnitude and direction in three dimensions, such as electric or magnetic field. A tensor of higher rank has magnitude and direction in more than three dimensions, such as curvature or torsion.

Tensors can be expressed in different coordinate systems, such as Cartesian, polar, cylindrical or spherical. To compare tensors in different coordinate systems, we need to use transformation rules that relate the components of tensors in one system to another. For example, if we have a vector $\vecv$ in Cartesian coordinates $(x,y,z)$ with components $v_x$, $v_y$ and $v_z$, we can transform it to polar coordinates $(r,\theta,\phi)$ with components $v_r$, $v_\theta$ and $v_\phi$ by using the following formulas:

$$ \beginaligned v_r &= v_x \cos \theta \sin \phi + v_y \sin \theta \sin \phi + v_z \cos \phi \\ v_\theta &= -v_x \sin \theta + v_y \cos \theta \\ v_\phi &= v_x \cos \theta \cos \phi + v_y \sin \theta \cos \phi - v_z \sin \phi \endaligned $$ Tensors can also be manipulated by using various operations, such as addition, subtraction, multiplication, contraction, differentiation and integration. For example, if we have two vectors $\veca$ and $\vecb$ with components $a_i$ and $b_i$ respectively ($i=1,2,\dots,n$), we can define their dot product as a scalar $c$ by using the following formula:

$$ c = \veca \cdot \vecb = a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n $$ We can also define their cross product as a vector $\vecc$ with components $c_i$ by using the following formula:

$$ \vecc = \veca \times \vecb = c_i = \epsilon_ijk a_j b_k $$ where $\epsilon_ijk$ is the Levi-Civita symbol that is equal to $+1$ if $(i,j,k)$ is an even permutation of $(1,2,3)$, $-1$ if it is an odd permutation, and $0$ if any two indices are equal.

What is Schaum's outline series?

Schaum's outline series is a collection of books that provide concise summaries of various topics in mathematics, science and engineering. They are designed to help students learn the essential concepts and techniques of each subject, and to prepare them for exams and assignments. Each book contains hundreds of solved problems with detailed explanations, as well as supplementary problems with answers for practice. The books are written by experts in their fields, and are updated regularly to reflect the latest developments and trends.

One of the books in the series is Schaum's outline of theory and problems of vector analysis and an introduction to tensor analysis by Seymour Lipschutz, Dennis Spellman and Murray R. Spiegel. This book covers the topics of vector analysis and tensor analysis in a clear and comprehensive way, with examples and applications from physics and engineering. It includes chapters on vectors and scalars, the dot and cross product, vector differentiation, gradient, divergence and curl, vector integration, the divergence theorem, Stokes' theorem and related integral theorems, curvilinear coordinates, tensor analysis, line element and metric tensor, associated tensors, Christoffel's symbols, length of a vector, angle between vectors, geodesics, covariant derivative, permutation symbols and tensors, tensor fields and operations.

Why is tensor analysis important for physics and engineering?

Tensor analysis is important for physics and engineering because it allows us to describe physical phenomena that involve multiple dimensions and coordinate systems in a concise and invariant way. For example, in classical mechanics, we can use tensors to express the laws of motion, such as Newton's second law $\vecF=m\veca$ or the conservation of momentum $\vecp=m\vecv$, in any coordinate system without changing their form. In fluid mechanics, we can use tensors to describe the flow of fluids, such as the continuity equation $\nabla \cdot \vecv=0$ or the Navier-Stokes equation $\rho (\frac\partial \vecv\partial t + \vecv \cdot \nabla \vecv) = -\nabla p + \mu \nabla^2 \vecv + \vecf$, where $\rho$ is the density, $p$ is the pressure, $\mu$ is the viscosity and $\vecf$ is the external force. In electromagnetism, we can use tensors to describe the electric and magnetic fields, such as Maxwell's equations $\nabla \cdot \vecE=\frac\rho\epsilon_0$, $\nabla \cdot \vecB=0$, $\nabla \times \vecE=-\frac\partial \vecB\partial t$ and $\nabla \times \vecB=\mu_0 (\vecJ + \epsilon_0 \frac\partial \vecE\partial t)$, where $\rho$ is the charge density, $\epsilon_0$ is the permittivity of free space, $\mu_0$ is the permeability of free space and $\vecJ$ is the current density. In relativity theory, we can use tensors to describe the curvature of space-time, such as Einstein's field equation $G_\mu\nu=\frac8\pi Gc^4 T_\mu\nu$ or the geodesic equation $\fracd^2 x^\mud s^2 + \Gamma^\mu_\alpha\beta \fracd x^\alphad s \fracd x^\betad s=0$, where $G_\mu\nu$ is the Einstein tensor, $T_\mu\nu$ is the stress-energy tensor, $G$ is the gravitational constant, $c$ is the speed of light, $\Gamma^\mu_\alpha\beta$ are the Christoffel symbols and $s$ is the proper time.

Main body

Basic concepts of tensors

Vectors and scalars

A vector is a quantity that has both magnitude and direction in one dimension. For example, the displacement of a particle from point A to point B can be represented by a vector $\vecAB$ that has a length equal to the distance between A and B, and a direction pointing from A to B. A scalar is a quantity that has only magnitude and no direction. For example, the speed of a particle is a scalar that measures how fast it is moving, regardless of its direction.

We can represent vectors and scalars by using symbols or by using components. A symbol is a letter with an arrow on top, such as $\vecv$ or $\vecF$. A component is a number that measures the magnitude of the vector along a certain axis or direction, such as $v_x$ or $F_y$. We can use different coordinate systems to define the axes or directions, such as Cartesian, polar, cylindrical or spherical. For example, in Cartesian coordinates $(x,y,z)$, we can write a vector $\vecv$ as $\vecv=v_x \hati + v_y \hatj + v_z \hatk$, where $\hati$, $\hatj$ and $\hatk$ are unit vectors along the $x$, $y$ and $z$ axes respectively. In polar coordinates $(r,\theta,\phi)$, we can write a vector $\vecv$ as $\vecv=v_r \hatr + v_\theta \hat\theta + v_\phi \hat\phi$, where $\hatr$, $\hat\theta$ and $\hat\phi$ are unit vectors along the radial, azimuthal and polar directions respectively.

Dot and cross product

The dot product (or scalar product) of two vectors is a scalar that measures the projection of one vector onto another. For example, if we have two vectors $\veca$ and $\vecb$ with an angle $\theta$ between them, we can define their dot product as $c=\veca \cdot \vecb=\veca \vecb \cos \theta$, where $\veca$ and $\vecb$ are the magnitudes of $\veca$ and $\vecb$ respectively. The dot product has some properties, such as commutativity ($\veca \cdot \vecb=\vecb \cdot \veca$), distributivity ($\veca \cdot (\vecb+\vecc)=\veca \cdot \vecb+\veca \cdot \vecc$) and linearity ($(\alpha \veca) \cdot (\beta \vecb)=\alpha \beta (\veca \cdot \vecb)$, where $\alpha$ and $\beta$ are scalars).

a})$, distributivity ($\veca \times (\vecb+\vecc)=\veca \times \vecb+\veca \times \vecc$) and linearity ($(\alpha \veca) \times (\beta \vecb)=\alpha \beta (\veca \times \vecb)$, where $\alpha$ and $\beta$ are scalars).

Vector differentiation

b})$ and $\fracddt(\veca \times \vecb)=\fracd\vecadt \times \vecb+\veca \times \fracd\vecbdt$) and chain rule ($\fracddt\vecr(u(t))=\fracdudt\fracd\vecrdu$, where $u(t)$ is a scalar function).

Gradient, divergence and curl

The gradient, divergence and curl are three important operators that can be applied to scalar or vector fields. A scalar field is a function that assigns a scalar value to each point in space, such as the temperature or the potential. A vector field is a function that assigns a vector value to each point in space, such as the velocity or the force.

The gradient of a scalar field is a vector field that points in the direction of the maximum rate of change of the scalar field. For example, if we have a scalar field $f(x,y,z)$, we can define its gradient as $\nabla f=\frac\partial f\partial x \hati + \frac\partial f\partial y \hatj + \frac\partial f\partial z \hatk$, where $\nabla$ is the del operator and $\hati$, $\hatj$ and $\hatk$ are unit vectors along the $x$, $y$ and $z$ axes respectively. The gradient has some properties, such as linearity ($\nabla (\alpha f+\beta g)=\alpha \nabla f+\beta \nabla g$, where $\alpha$ and $\beta$ are scalars), product rule ($\nabla (fg)=f\nabla g+g\nabla f$) and chain rule ($\nabla f(g(x,y,z))=\nabla g(x,y,z) \cdot \nabla f(g(x,y,z))$, where $g(x,y,z)$ is a scalar function).

\vecF) dV=\oint_S \vecF \cdot d\vecS$, where $V$ is a volume and $S$ is its boundary surface).

The curl of a vector field is a vector field that measures the tendency of the vector field to rotate around each point. For example, if we have a vector field $\vecF(x,y,z)$, we can define its curl as $\nabla \times \vecF=(\frac\partial F_z\partial y-\frac\partial F_y\partial z) \hati + (\frac\partial F_x\partial z-\frac\partial F_z\partial x) \hatj + (\frac\partial F_y\partial x-\frac\partial F_x\partial y) \hatk$, where $F_x$, $F_y$ and $F_z$ are the components of $\vecF$ along the $x$, $y$ and $z$ axes respectively. The curl has some properties, such as anti-linearity ($\nabla \times (\alpha \vecF+\beta \vecG)=\alpha \nabla \times \vecF-\beta \nabla \times \vecG$, where $\alpha$ and $\beta$ are scalars), product rule ($\nabla \times (f\vecF)=f\nabla \times \vecF-\vecF \times \nabla f$) and Stokes' theorem ($\int_S (\nabla \times \vecF) \cdot d\vecS=\oint_C \vecF \cdot d\vecr$, where $S$ is a surface and $C$ is its boundary curve).

Advanced topics of tensors

Vector integration

r}\partial u \times \frac\partial \vecr\partial v) dudv$, which measures the flux of $\vecF$ through $S$. If we have a volume $V$ bounded by a surface $S$, we can define the volume integral of $\vecF$ over $V$ as $\int_V \vecF \cdot d\vecV=\iiint_V \vecF(x,y,z) \cdot dxdydz$, which measures the total amount of $\vecF$ in $V$. Vector integration has some theorems, such as the divergence theorem ($\int_V (\nabla \cdot \vecF) dV=\oint_S \vecF \cdot d\vecS$), Stokes' theorem ($\int_S (\nabla \times \vecF) \cdot d\vecS=\oint_C \vecF \cdot d\vecr$) and Green's theorem ($\oint_C (Pdx+Qdy)=\iint_D (\frac\partial Q\partial x-\frac\partial P\partial y)dxdy$, where $P$ and $Q$ are scalar functions).

Divergence theorem, Stokes' theorem and related integral theorems

The divergence theorem, Stokes' theorem and related integral theorems are generalizations of the fundamental theorem of calculus that relate the integrals of a vector field over different types of regions. The fundamental theorem of calculus states that $\int_a^b f'(x) dx=f(b)-f(a)$, where $f(x)$ is a scalar function. This means that the integral of the derivative of a function over an interval is equal to the difference of the function at the endpoints of the interval.

The divergence theorem states that $\int_V (\nabla \cdot \vecF) dV=\oint_S \vecF \cdot d\vecS$, where $\vecF(x,y,z)$ is a vector field, $V$ is a volume and $S$ is its boundary surface. This means that the integral of the divergence of a vector field over a volume is equal to the integral of the vector field over its boundary surface. The divergence theorem can be used to calculate the net flux of a vector field through a closed surface.

Stokes' theorem states that $\int_S (\nabla \times \vecF) \cdot d\vecS=\oint_C \vecF \cdot d\vecr$, where $\vecF(x,y,z)$ is a vector field, $S$ is a surface and $C$ is its boundary curve. This means that the integral of the curl of a vector field over a surface is equal to the integral of the vector field over its boundary curve. Stokes' theorem can be used to calculate the circulation of a vector field around a closed curve.

E(x,y,z)$ is the electric field, $S$ is a closed surface and $Q$ is the total charge enclosed by $S$) and Ampere's theorem ($\oint_C (\vecB \cdot d\vecr)=\mu_0 I$, where $\vecB(x,y,z)$ is the magnetic field, $C$ is a closed curve and $I$ is the total current passing through $C$).

Curvilinear coordinates

Curvilinear coordinates are coordinate systems that use curved lines or surfaces to define the position of a point in space. For example, polar coordinates $(r,\theta)$ use a radial line and an angular line to locate a point in the $xy$-plane, cylindrical coordinates $(r,\theta,z)$ use a radial line, an angular line and a vertical line to locate a point in the $xyz$-space, and spherical coordinates $(r,\theta,\phi)$ use a radial line, an azimuthal angle and a polar angle to locate a point in the $xyz$-space. Curvilinear coordinates are useful for describing physical phenomena that have symmetry or periodicity in certain directions.

To use curvilinear coordinates, we need to know how to convert them to Cartesian coordinates and vice versa. For example, to convert from polar coordinates $(r,\theta)$ to Cartesian coordinates $(x,y)$, we can use the following formulas:

$$ \beginaligned x &= r \cos \theta \\ y &= r \sin \theta \endaligned $$ To convert from Cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$, we can use the following formulas:

$$ \beginaligned r &= \sqrtx^2+y^2 \\ \theta &= \tan^-1 \fracyx \endaligned $$ We also need to know how to calculate the length of a curve, the area of a surface and the volume of a region in curvilinear coordinates. For example, to calculate the length of a curve $C$ parametrized by $\vecr(t)=r(t) \hatr + \theta(t) \hat\theta$ for $a \leq t \leq b$ in polar coordinates, we can use the following formula:

$$ L=\int_C ds=\int_a^b \fracd\vecrdt dt=\int_a^b \sqrt(r')^2+r^2(\theta')^2 dt $$ where $r'$ and $\theta'$ are the derivatives of $r(t)$ and $\theta(t)$ respectively. To calculate the area of a surface $S$ parametrized by $\vecr(u,v)=r(u,v) \hatr + \theta(u,v) \hat\theta$ for $(u,v) \in D$ in polar coordinates, we can use the following formula:

$$ A=\int_S dS=\iint_D \frac\partial \vecr\partial u \times \frac\partial \vecr\partial v dudv=\iint_D r(u,v) \frac\partial (r,\theta)\partial (u,v) dudv $$ where $\frac\partial (r,\theta)\partial (u,v)$ is the Jacobian determinant of the transformation from $(u,v)$ to $(r,\theta)$. To calculate the volume of a region $V$ bounded by a surface $S$ in polar coordinates, we can use the following formula:

$$ V=\int_V dV=\iint_S r dS=\iint_D r^2 \frac\partial (r,\theta)\partial (u,v) dudv $$ Tensor fields and operations

A tensor field is a function that assigns a tensor value to each point in space. For example, a scalar field is a tensor field of rank zero that assigns a scalar value to each point in space, such as the temperature or the potential. A vector field is a tensor field of rank one that assigns a vector value to each point in space, such as the velocity or the force. A matrix field is a tensor field of rank two that assigns a matrix value to each point in space, such as the stress or the strain. A tensor field of higher rank assigns a higher-order tensor value to each point in space, such as the curvature or the torsion.

Tensor fields can be expressed in different coordinate systems, such as Cartesian, polar, cylindrical or spherical. To compare tensor fields in different coordinate systems, we need to use transformation rules that relate the components of tensor fields in one system to another. For example, if we have a vector field $\vecv$ in Cartesian coordinates $(x,y,z)$ with components $v_x$, $v_y$ and $v_z$, we can transform it to polar coordinates $(r,\theta,\phi)$ with components $v_r$, $v_\theta$ and $v_\phi$ by using the same formulas as before:

$$ \beginaligned v_r &= v_x \cos \theta \sin \phi + v_y \sin \theta \sin \phi + v_z \cos \phi \\ v_\theta &= -v_x \sin \theta + v_y \cos \theta \\ v_\phi &= v_x \cos \theta \cos \phi + v_y \sin \theta \cos \phi - v_z \sin \phi \endaligned $$ If we have a matrix field $A$ in Cartesian coordinates $(x,y,z)$ with components $A_ij$ ($i,j=1,2,3$), we can transform it to polar coordinates $(r,\theta,\phi)$ with components $A_\alpha\beta$ ($\alpha,\beta=r,\theta,\phi$) by using the following formula:

$$ A_\alpha\beta=\frac\partial x^i\partial x^\alpha \frac\partial x^j\partial x^\beta A_ij $$ where $\frac\partial x^i\partial x^\alpha$ and $\frac\partial x^j\partial x^\beta$ are the partial derivatives of the Cartesian coordinates with respect to the polar coordinates. For example, $\frac\partial x\partial r=\cos \theta \sin \phi$, $\frac\partial y\partial r=\sin \theta \sin \phi$ and $\frac\partial z\partial r=\cos \phi$.

Tensor fields can also be manipulated by using various operations, such as addition, subtraction, multiplication, contraction, differentiation and integration. For example, if we have two vector fields $\veca$ and $\vecb$ with components $a_i$ and $b_i$ respectively ($i=1,2,\dots,n$), we can define their dot product as a scalar field $c$ by using the same formula as before:

$$ c = \veca \cdot \vecb = a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n $$ We can also define their cross product as a vector field $\vecc$ with components $c_i$ by using the same formula as before:

$$ \vecc = \veca \times \vecb = c_i = \epsilon_ijk a_j b_k $$ where $\epsilon_ijk$ is the Levi-Civita symbol that is equal to $+1$ if $(i,j,k)$ is an even permutation of $(1,2,3)$, $-1$ if it is an odd permutation, and $0$ if any two indices are equal.

Conclusion

Summary of the main points

In this article, we have introduced the basic concepts and applications of tensor analysis, and showed you how to download the pdf file of a popular book on this topic: Schaum's outline of theory and problems of vector analysis and an introduction to tensor analysis by Seymour Lipschutz, Dennis Spellman and Murray R. Spiegel. We have covered the topics of vectors and scalars, the dot and cross product, vector differentiation, gradient, divergence and curl, vector integration, the divergence theorem, Stokes' theorem and related integral theorems, curvilinear coordinates, tensor analysis, line element and metric tensor, associated tensors, Christoffel's symbols, length of a vector, angle between vectors, geodesics, covariant derivative, permutation symbols and tensors, tensor fields and operations.

Benefits of learning tensor analysis wi