Gradient Theorem

6 Key excerpts on "Gradient Theorem"

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  971 15 We begin this chapter by introducing the concept of a vector field, an important tool for the study of gravitational and electrostatic force fields, the flow of fluids, and conservation of energy. Next, we will introduce the “line integral,” a new type of integral with a variety of applications to the analysis of vector fields. Finally, we conclude with three major theorems, Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem. These theorems provide deep insight into the nature of vector fields and are the basis for many of the most important principles in physics and engineering. TOPICS IN VECTOR CALCULUS 15.1 VECTOR FIELDS In this section, we will consider functions that associate vectors with points in 2-space or 3-space. We will see that such functions play an important role in the study of fluid flow, gravitational force fields, electromagnetic force fields, and a wide range of other applied problems. VECTOR FIELDS Consider a unit point-mass located at any point in the Universe. According to Newton’s Law of Universal Gravitation, the Earth exerts an attractive force on the mass that is directed toward the center of the Earth and has a magnitude that is inversely proportional to the square of the distance from the mass to the Earth’s center (Figure 15.1.1). This association of force vectors with points in space is called the Earth’s gravitational field. A similar Figure 15.1.1 association occurs in fluid flow. Imagine a stream in which the water flows horizontally at every level, and consider the layer of water at a specific depth. At each point of the layer, the water has a certain velocity, which we can represent by a vector at that point (Figure 15.1.2). This association of velocity vectors with points in the two-dimensional layer is called the velocity field at that layer. These ideas are captured in the following definition.

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  Foundations of Classical Mechanics

  • P. C. Deshmukh(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  Gradient Operator, Methods of Fluid Mechanics, and Electrodynamics CHAPTER 10 There is no greater burden than an unfulfilled potential. —Charles M. Schulz 10.1 THE SCALAR FIELD, DIRECTIONAL DERIVATIVE, AND GRADIENT In the discussion on Fig. 2.4 (Chapter 2), we learned that it was neither innocuous to define a vector merely as a quantity that has both direction and magnitude, nor to define a scalar simply as a quantity that has magnitude alone. It is not that the properties referred here of a scalar and a vector are invalid. Rather, it is to be understood that these properties do not provide an unambiguous definition. Only a signature criterion of a physical quantity can be used to define it. We therefore introduced, in Chapter 2, comprehensive definitions of the scalar as a tensor of rank 0, and of the vector as a tensor of rank 1. In this chapter we shall acquaint ourselves with the mathematical framework in which the laws of fluid mechanics and electrodynamics are formulated using vector algebra and vector calculus. In fact, the techniques are used not merely in these two important branches of classical mechanics, but also in very various other subdivisions of physics. The background material seems at times to be intensely mathematical, but that is only because the laws of nature engage a mathematical formulation very intimately, as we encounter repeatedly in the analysis of physical phenomena. There are many excellent books in college libraries from which one can master the mathematical methods. These topics are extremely enjoyable to learn; they help us develop rigorous insights in the laws of nature. The literature on these topics is vast. A couple of illustrative books [1, 2] are suggested for further reading. We consider the example of a particular scalar function, namely the temperature distribution in a room. The temperature is a physical property at a particular point in space, such as the point P in Fig.

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  Mathematical Methods for Mathematicians, Physical Scientists and Engineers

  • Jeremy Dunning-Davies(Author)
  • 2003(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Chapter 10 Vector Analysis 10.1 GRADIENT OF A SCALAR FIELD The concept of 'rate of change' of a function is fundamental in differential calculus. To discuss the change of a scalar field Φ ( Γ ) , in a region of three-dimensional space, a vector field—known as the gradient of Φ (Γ )—may be de-fined throughout the region. Actually, since the magnitude of a scalar field is a function of position, it is quite natural to seek geometrical interpretations of the derivatives of the magnitude. Suppose u x , u 2 , u 3 are three coordinates in space. Then, the position vector r will be a function of u u u 2 , u 3 . It follows that φ will be a function of i» 2 , «3, also: φ = φ (ΐί!, u 2 , u 3 ). If φ is differentiable, θφ Φ ( Γ + 6r) = Φ (Γ ) + — δίΛ + higher-order terms. du, Here 6r is the displacement arising from changes δΐί ( in the three coordinates Uj. In general, bu t will not be the components of 8r. However, Cartesian coor-dinates Xi are defined to be the Cartesian components of r and so δχ ; are the Cartesian components of 5r. Thus, Cartesian coordinates simplify the work considerably. Hence dφ = Φ ( Γ +6r) -Φ ( Γ ) = νφ·δΓ = higher-order terms where [νΦ1; = θφ/θχ,. Thus, the gradient of the scalar field Φ ( Γ ) , denoted by grad φ or νψ (del φ), is defined to be the vector whose Cartesian components are θφ/θχ;. The gradient of φ may be written also in the form θφ θφ . θ φ . νψ = — i + — j + — k dx dy dz Sec. 10.1] Gradient of a Scalar Field 267 where i, j , k are the usual basis vectors, which have been encountered fre-quently already. However, in terms of the orthogonal curvilinear coordinates u, u 2 , it 3 , θφ θφ θφ dφ = — d«i + — du 2 + — 0U3 dUi du 2 du 3 1 θφ 1 θφ 1 θφ — — · (Aidax, A 2 d« 2 , h 3 du 3 ).

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  Mathematical Methods for Optical Physics and Engineering

  • Gregory J. Gbur(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  2.3 The gradient, ∇ The gradient is an operation upon a scalar field which results in a vector field: if φ(x, y, z) is a scalar field defined in three-dimensional space, ∇φ(x, y, z) is a vector field. In Cartesian coordinates, it has the form ∇φ(x, y, z) ≡ ∂φ ∂ x ˆ x + ∂φ ∂ y ˆ y + ∂φ ∂ z ˆ z. (2.30) It is to be recalled from Section 1.2 that a vector must satisfy specific transformation properties under a rotation of coordinates; we will show that this is the case for ∇φ at the end of this section. To understand the significance of the gradient, we first consider a one-dimensional scalar function f (t ). We know from elementary calculus that the instantaneous rate of change of this function is given by its derivative, df (t )/dt . Furthermore, as t is changed by an infinitesimal amount dt , the function f (t ) changes by an amount df = [df (t )/dt ]dt . Let us now consider a scalar field φ(x, y, z) defined in three-dimensional space, and we consider a path through that three-dimensional space characterized by s(t ), where t is a real parameter which varies from 0 to 1 – that is, as t increases from 0 to 1, our position on the path changes continuously from s 0 to s 1 . This formulation is the same used in Section 2.2.1, and was illustrated in Fig. 2.1. Let us investigate how the function φ changes as we move along the path s, i.e. as we move an infinitesimal distance ds along the path. We can partly characterize this by determining 36 Vector calculus the derivative of φ with respect to the parameter t , i.e. dφ[x(t ), y(t ), z(t )]/dt . Using the chain rule, we can write this path derivative in terms of Cartesian coordinates as dφ dt = ∂φ ∂ x dx dt + ∂φ ∂ y dy dt + ∂φ ∂ z dz dt . (2.31) It is to be noted that this derivative is taken with respect to the dimensionless parameter t ; we now need to use this formula to find a true spatial derivative of the function φ, i.e.

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  Calculus

  Late Transcendental

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  971 15 Results in this chapter provide tools for analyzing and understanding the behavior of hurricanes and other fluid flows. TOPICS IN VECTOR CALCULUS We begin this chapter by introducing the concept of a vector field, an important tool for the study of gravitational and electrostatic force fields, the flow of fluids, and conservation of energy. Next, we will introduce the “line integral,” a new type of integral with a variety of applications to the analysis of vector fields. Finally, we conclude with three major theorems, Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem. These theorems provide deep insight into the nature of vector fields and are the basis for many of the most important principles in physics and engineering. 15.1 VECTOR FIELDS In this section we will consider functions that associate vectors with points in 2-space or 3-space. We will see that such functions play an important role in the study of fluid flow, gravitational force fields, electromagnetic force fields, and a wide range of other applied problems. VECTOR FIELDS Consider a unit point-mass located at any point in the Universe. According to Newton’s Law of Universal Gravitation, the Earth exerts an attractive force on the mass that is di- rected toward the center of the Earth and has a magnitude that is inversely proportional to the square of the distance from the mass to the Earth’s center (Figure 15.1.1). This associa- tion of force vectors with points in space is called the Earth’s gravitational field. A similar Figure 15.1.1 association occurs in fluid flow. Imagine a stream in which the water flows horizontally at every level, and consider the layer of water at a specific depth. At each point of the layer, the water has a certain velocity, which we can represent by a vector at that point (Figure 15.1.2). This association of velocity vectors with points in the two-dimensional layer is called the velocity field at that layer.

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  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  The Fundamental Theorem of Line Integrals One version of the Fundamental Theorem of Calculus states that Z b a f 0 (x) dx = f (b) - f (a) In words, integrating the rate of change of a function over an interval gives the net change of the function, the difference between its ending and starting values. The Fundamental Theorem of Line Integrals generalizes this result to functions defined on curves. To account for how the curve turns, we integrate the rate of change of f in the direction of the tangent vector to obtain the net change of the function. Recall that the rate of change of f in the direction of a unit vector u is given by the directional derivative D u f = ∇f · u. So the rate of change of f in the direction of the unit tangent vector T at a point (x, y, z) is given by ∇f (x, y, z) · T(x, y, z) Now if we integrate this function over a curve C beginning at P and ending at Q, we have Net change of f on C = Z C ∇f (x, y, z) · T(x, y, z) ds = f (Q) - f (P ) If C has a parameterization r(t) for a ≤ t ≤ b, then Z C ∇f (x, y, z) · T(x, y, z) ds = Z b a ∇f ( r(t) ) · T(t) kr 0 (t)k dt = Z b a ∇f ( r(t) ) · r 0 (t) kr 0 (t)k kr 0 (t)k dt = Z b a ∇f ( r(t) ) · r 0 (t) dt = Z C ∇f · dr 898 CHAPTER 15. VECTOR ANALYSIS This informal line of reasoning has led us to the Fundamental Theorem of Line Integrals. Theorem 15.3.1 The Fundamental Theorem of Line Integrals Let f be a differentiable function with a continuous gradient and let C be a smooth (or piecewise smooth) curve with parameterization r(t) for a ≤ t ≤ b. Then Z C ∇f ·dr = f ( r(b) ) -f ( r(a) ) = f (ending point of C)-f (beginning point of C) In words, the line integral of a conservative vector field over a curve is equal to the difference in the values of the potential function at the endpoints of the curve. Proof For convenience, we prove the theorem in two dimensions. The proof extends naturally to three or more dimensions.

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