Vectors in Calculus

7 Key excerpts on "Vectors in Calculus"

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  Vector Calculus

  • William Cox(Author)
  • 1998(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  8.1 Introduction: what is a vector? You are probably aware from your previous mathematics that one of the most powerful tools for dealing with functions of more than one variable is the use of vector notation. Thus, when describing the motion of a projectile in two dimen-sions we have the option of using (x(t), yet)) coordinates to represent position at any time or of using the position vector ret) = x(t)i + y(t)j with i, j the usual basis vectors. The point about using vectors here is that it effec-tively reduces the number of 'variables' -instead of two coordinates, we have one vector. A natural question therefore is to what extent one can use vectors in the theory of functions of several variables -which is what this book is about. So far, we have managed virtually without them -and indeed it is not always clear that they would be of much use. But in fact they are -the bulk of the rest of this book is essentially about 'vector calculus'. Why should this be so? Why does most of the differen-tiation and integration that we cover occur in a vector context? Is it just the case that vectors are simply a good shorthand notation whenever we have more than one variable? No, there is a deep significance to the idea of a vector which is simply not brought out in the introductory treatments to which you may have been exposed so far. Vectors are of fundamental importance and utility in all physical applications -they are far more than simply a shorthand notation. The reason for this is rather subtle, and requires a leap into abstraction which really has to be deferred until sufficient mathematical foundations have been laid -that is why elementary treatments are usually incomplete. To get an insight into the new view-point, let us stick with our two-dimensional projectile. You may have a number of different definitions, but they are most likely to come from the following list: 1. Any quantity having both a magnitude and a direction, such as velocity as opposed to speed.

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  Mathematical Methods in the Earth and Environmental Sciences

  • Adrian Burd(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 Vectors and Calculus We have seen that many environmental quantities we are interested in can be described mathematically using vectors. To understand how these vectors change spatially and temporally we need to combine what we know of vectors with our knowledge of calculus— a subject creatively called vector calculus. In this chapter we will largely be dealing with vector fields, where there is a vector (a fluid velocity, for example) that has a value at every point in the space we are considering (e.g., the ocean, the atmosphere, or a lake). This means that a vector field (F) is a function of the spatial coordinates, F = F( x , y), so that as x and y change, so do the magnitude and the direction of the vector. Some examples of vector fields are shown in Figure 7.1. In this chapter we will learn to calculate how these vectors change as the values of x , y, and z change (the “calculus” of vector calculus) and discover some useful theorems that connect integrals of vectors along paths and over surfaces. Vector calculus lies at the heart of understanding many processes in the Earth and environmental sciences. To start with, it provides a framework for describing how moving fluids transport material in the environment and how heat and chemical compounds diffuse. For example, it provides a mathematical framework for understanding how heat moves from the Earth’s core to the surface, how pollutants are transported through groundwater flows, how vortices are formed and move, how the Earth’s magnetic field behaves, how chemicals move through the natural environment, and a myriad other natural processes. This makes vector calculus a very powerful tool. 7.1 Differentiating a Vector Let us start by thinking about how we differentiate a vector. The basic idea is very similar to that of differentiating a function, except that we have the added complication that a vector can be described in terms of components and basis vectors.

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  Essential Mathematical Methods for the Physical Sciences

  • K. F. Riley, M. P. Hobson(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Vector calculus In Section A.9 of Appendix A we review the algebra of vectors, and in Chapter 1 we considered how to transform one vector into another using a linear operator. In this chapter and the next we discuss the calculus of vectors, i.e. the differentiation and integration both of vectors describing particular bodies, such as the velocity of a particle, and of vector fields, in which a vector is defined as a function of the coordinates throughout some volume (one-, two-or three-dimensional). Since the aim of this chapter is to develop methods for handling multi-dimensional physical situations, we will assume throughout that the functions with which we have to deal have sufficiently amenable mathematical properties, in particular that they are continuous and differentiable. 2.1 Differentiation of vectors • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Let us consider a vector a that is a function of a scalar variable u . By this we mean that with each value of u we associate a vector a ( u ). For example, in Cartesian coordinates a ( u ) = a x ( u ) i + a y ( u ) j + a z ( u ) k , where a x ( u ), a y ( u ) and a z ( u ) are scalar functions of u and are the components of the vector a ( u ) in the x -, y -and z -directions respectively. We note that if a ( u ) is continuous at some point u = u 0 then this implies that each of the Cartesian components a x ( u ), a y ( u ) and a z ( u ) is also continuous there. Let us consider the derivative of the vector function a ( u ) with respect to u . The derivative of a vector function is defined in a similar manner to the ordinary derivative of a scalar function f ( x ). The small change in the vector a ( u ) resulting from a small change u in the value of u is given by a = a ( u + u ) − a ( u ) (see Figure 2.1 ).

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  Elementary Mathematical and Computational Tools for Electrical and Computer Engineers Using MATLAB

  • Jamal T. Manassah(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  7 Vectors

  In this chapter, we consider both finite- and infinite-dimensional vectors. Using the Dirac notation, we show that the two cases can be conceptually treated in similar fashions. Very basic results from vector calculus in 2-D are used to solve the problem of planetary motion. In the last section, the Fourier series and the expansion in Legendre polynomials are used to illustrate the expansion in particular basis functions in infinite-dimensional spaces.

  7.1 Vectors in Two Dimensions

  A vector in 2-D is defined by its length and the angle it makes with a reference axis (usually the x -axis). This vector is represented graphically by an arrow. The tail of the arrow is called the initial point of the vector and the tip of the arrow is the terminal point. Two vectors are equal when both their length and angle with a reference axis are equal.

  7.1.1 Addition

  The sum of two vectors

  u →

  +

  v →

  =

  w →

  is a vector constructed graphically as follows. At the tip of the first vector, draw a vector equal to the second vector, such that its tail coincides with the tip of the first vector. The resultant vector has as its tail that of the first vector, and as its tip, the tip of the just-drawn second vector (the Parallelogram rule) (see Figure 7.1 ).

  The negative of a vector is that vector whose tip and tail have been exchanged from those of the vector. This leads to the conclusion that the difference of two vectors is the other diagonal in the parallelogram (Figure 7.2 ).

  7.1.2 Multiplication of a Vector by a Real Number

  If we multiply a vector

  v →

  by a real number k , the result is a vector whose length is k times the length of

  v →

  , and whose direction is that of

  v →

  if k is positive, and opposite if k is negative.

  FIGURE 7.1 Sum of two vectors.

  FIGURE 7.2 Difference of two vectors.

  7.1.3 Cartesian Representation

  It is most convenient for a vector to be described by its projections on the x -axis and on the y -axis, respectively; these are denoted by (v 1 , v 2 ) or (

  vx

  ,

  vy

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  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R 2 Vector Calculus 2.1 Introduction The application of vector methods to physical problems most frequently takes the form of differential operations. The rate of change of a vector function with respect to the spatial coordinates or with respect to the time are of particular importance. Such operations allow us, for example, to define the velocity vector of the motion of a particle or to describe the flow properties of a fluid. In this chapter we begin by defining the elemen-tary differential operations which immediately allow us to calculate the velocity and acceleration vectors in the commonly used coordinate systems. Angular velocity is considered next and this leads to a discussion of infinitesimal rotations. Treated next is the important differential operator, the gradient. The fact that the gradient operator may act on vector functions in different ways, leads finally to the divergence and the curl. The chapter concludes with a brief discussion of the simple integral concepts that are necessary in mechanics. 32 2.2 DIFFERENTIATION OF A VECTOR WITH RESPECT TO A SCALAR 33 2.2 Differentiation of a Vector with Respect to a Scalar If a scalar function φ = cp(s) is differentiated with respect to the scalar variable s, then since neither part of the derivative can change under a coordinate transformation, the derivative itself cannot change and must therefore be a scalar. That is, in the x f and x coordinate systems, φ = φ' and s — s', so that άφ = άφ' and ds = ds'. Hence, dcp/ds = d(p'/ds' = (άφ/ds)' (2.1) In a similar manner, we may formally define the differentiation of a vector A with respect to a scalar s.

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  Essential Mathematics for NMR and MRI Spectroscopists

  • Keith C Brown(Author)
  • 2020(Publication Date)
  • Royal Society of Chemistry

    (Publisher)

  4 Vector Calculus

  Mathematics is the queen of the sciences and arithmetic is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to first rank.

  Carl Friedrich Gauss

  4.1 Vector Differentiation

  In order to introduce electric and magnetic field theories and, ultimately, the rationale for various parts of the NMR Hamiltonian we must review our vector calculus.†

  Vector differentiation (with respect to time, which is what we will ultimately be interested in) is defined for a static coordinate system by:

  (4.1)

  and for a dynamic coordinate system by:

  (4.2)

  for a rotating coordinate system in which the unit vectors are changing direction with time. Vector differentiation follows much the same rules for normal function differentiation:

  For the orthogonal Cartesian unit vectors:

  What direction is d x /dt in? Since x is a unit vector, d x /dt cannot change in magnitude. The only other change that is possible is a change in direction … a rotation in other words. It is easiest to picture in the situation in which the rotation is about the z -axis (see Figure 4.1 ).

  Figure 4.1 The rotating unit vector, .

  So, d x /dt is at right angles to x along the y -axis and we can write:

  Why x ·b x ? Since d y /dt must be at right angles to y , it must be parallel and proportional to x . The same reasoning applies to y ·a y .

  The vector operator, , is often referred to as del or nabla and is defined as:

  (4.3)

  The gradient is defined in terms of del and a scalar function, f (x, y, z ):

  (4.4)

  f (x, y, z ) is a scalar function that defines a scalar field and the action of on f (x , y , z ) produces a vector function (sometimes referred to as a vector-valued function). eqn (4.4) contains three components of the vector function, P , Q and R . A scalar field is a region of space in which there is a scalar value associated with each point in the space. For example, one might define a scalar field of temperatures within a region of the atmosphere. Similarly, a vector field

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  14 CALCULUS OF VECTOR-VALUED FUNCTIONS 14.1 Vector-valued functions of a real variable This chapter combines vector algebra with the methods of calculus and describes some appli- cations to the study of curves and to some problems in mechanics. The concept of a vector-valued function is fundamental in this study. definition. A function whose domain is a set of real numbers and whose range is a subset of n-space V n is called a vector-valued function of a real variable. We have encountered such functions in Chapter 13. For example, the line through a point P parallel to a nonzero vector A is the range of the vector-valued function X given by X(t) = P + tA for all real t. Vector-valued functions will be denoted by capital letters such as F, G, X, Y , etc., or by small bold-face italic letters f , g, etc. The value of a function F at t is denoted, as usual, by F(t). In the examples we shall study, the domain of F will be an interval which may contain one or both endpoints or which may be infinite. 14.2 Algebraic operations. Components The usual operations of vector algebra can be applied to combine two vector-valued functions or to combine a vector-valued function with a real-valued function. If F and G are vector-valued functions, and if u is a real-valued function, all having a common domain, we define new functions F + G, uF, and F ⋅ G by the equations (F + G)(t) = F(t) + G(t), (uF)(t) = u(t)F(t), (F . G)(t) = F(t) . G(t). The sum F + G and the product uF are vector valued, whereas the dot product F ⋅ G is real valued. If F(t) and G(t) are in 3-space, we can also define the cross product F × G by the formula (F × G)(t) = F(t) × G(t). 512 Limits, derivatives, and integrals 513 The operation of composition may be applied to combine vector-valued functions with real-valued functions.

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