Video created by the hong kong university of science and technology for the course vector calculus for engineers. Scalars may or may not have units associated with them. Vector field, which is a gradient of a scalar field fx, y, z is irrotational, and. In part iii of this book we shall see how to associate a form gu to a vector. Formally, scalar is a word used to distinguish the field from a vector field.
A vector function defines a vector field and a scalar function defines a scalar. We usually need to compute the value of the scalar quantity, its gradient, and. To avoid this ambiguity, we use term linear space for the vector. Its a function of x and y, and its equal to some scalar function of x and y times the iunit vector, or the horizontal unit vector, plus some other function, scalar. They are fancy words for functions usually in context of differential equations. Scalar field and vector field physics video in hindi. The divergence takes a vector field and gives back a scalar field. Conservative vector fields recall the diagram we drew last week depicting the derivatives weve learned in the 32 sequence. The signs of the free field terms s, and sp are fixed by the requirement that the corresponding free. Pdf copies of these notes including larger print versions, tutorial sheets, faqs etc will be accessible from. Scalar fields, vector fields and covector fields first we study scalar. Discovery exercise for scalar and vector fields the xyplane is covered with sand.
Vectors are quantities that are fully described by both a magnitude and a direction. Learn vector calculus for engineers from the hong kong university of science and technology. The force is a vector field, which can be obtained as a factor of the gradient of the potential energy scalar field. The gradient of a scalar field is a vector field and whose magnitude is the rate of change and which points in the direction of the greatest rate of increase of the scalar field. A scalar field is a function that gives us a single value of some variable for every. The gradient of a scalar field provides a vector field that states how the scalar.
For a layperson, the two terms, are same, but in the world of physics, there is a huge difference between scalar and vector. Scalar fields, vector fields and covector fields scalar. Scalar and vector fields learn about education and b. Scalar and vector fields lecture 9 vector calculus for. In physical problems dealing with either scalar or vector fields. Here is an example of the visualization of a vector. Vectors are used to describe physical quantities such as velocity, momentum, acceleration and force. The subject of chapter 10 is the representation of field operators in orthogonal curvilinear coordinate systems.
The vector space of all 1forms is called v sometimes it is called the dual space of v. If the vector is resolved, its components represent the rate of change of the scalar field. A particle a of mass m is fixed to a point p0 and a particle b of mass m is free. It is important to note that the use of the dot in this context is not meant to say that this is the inner product scalar product of two vectors. These quantities are often described as being a scalar or a vector quantity. Wandering the plane in your bare feet, you conclude that the depth of sand at any given point can be modeled by the equation.
The remainder of this lesson will focus on several examples of vector and scalar. A vector field for which the curl vanishes is said to be an irrotational field, or conservative field. The gradient of a scalar field provides a vector field that states how the scalar value is changing throughout spacea change that has both a magnitude and direction. Divergence is a scalar, that is, a single number, while curl is itself a vector.
The lagrangian for the free scalar field has the following form 5. A few examples of these include force, speed, velocity and work. But before that, it is suggested to cover basic linear algebra for clarification of possible ambiguity of vector. Recall that every continuous scalar function of a single variable is the derivative of some function. We know that many physical quantities like temperature, electric or gravitational field etc. A scalar function is a function that assigns a real number i. This chapter is concerned with applying calculus in the context of vector fields. Chapter 1 fields massachusetts institute of technology. In this article, learn what are scalar and vector fields. The user manual piolle and bentamy 2002 gives a comprehensive description of. Generation of scalable vector graphics svg files, which are easily editable and convertible while preserving the best quality for inclusion in a pdf. Scalar and vector functions, point functions, scalar point.
If we assign a vector which represents the wind velocity at that point, we obtain an example of a vector field. Given any field theory in lagrangian form, there exist standard procedures for converting it to hamiltonian form and for constructing the energymomentum tensor of the field. Vector field, which is a gradient of a scalar field. Pdf representation of divergencefree vector fields researchgate. Scalar field, vector field and level surface duration. Pdf this paper focuses on a representation result for divergencefree vector fields. A major thread of vector calculus investigates if this generalizes in some sense to vector. Scalars and vectors are differentiated depending on their definition. Scalar and vector fields lecture 9 vectors coursera. Scalars and vectors scalars and vectors a scalar is a number which expresses quantity. In vector analysis we deal with scalar and vector functions. The mathematical quantities explaining the motion of a body are bifurcated into two groups, i.
One distinction between a scalar field and the vector electromagnetic field is that the field. A lot of mathematical quantities are used in physics to explain the concepts clearly. Scalar fields takes a point in space and returns a number. This section will tell you how to define the scalar, spinor and vector tensor field. We know that many physical quantities like temperature, electric or gravitational field, etc. This is a very simple question,but any one is free to ask what he likes.
In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. In the first week we learn about scalar and vector fields, in the second week about differentiating fields. Displacement, velocity, acceleration, electric field. Missing data are handled and do not interfere with the flow lines. Difference between scalar and vector quantity with.
So lets say that i have a vector field f, and were going to think about what this means in a second. The divergence is a scalar measuring net flux of the field from each point. Let f be a scalar function, such that any point x is attached a real number. Scalar, spinor and vector tensor field path integral. Scalars are quantities that are fully described by a magnitude or numerical value alone. The laplacian may be applied to a vector field as well. In many situtations only the magnitude and direction of a. Vector fields takes a point in space and returns a vector. The integral properties of fields, stokes, gausss, and greens theorems are the subjects of chapter 9. A particle a of mass m is fixed to a point p0 and a particle b of mass m is free to take up various positions p in space. Any scalar is a quantity number without direction,like speed,temperature,while the vector has quantity and direction,like. Ordered pair f is called a stationary scalar field, while function f is called the potential.
In physical problems dealing with either scalar or vector fields it is important to know. What is the difference between scalar field and vector. Visualizing twodimensional scalar and vector field. Rotational field contains whirls at those points, at which curl is nonzero, and it determines the direction of a flow at this point. Construct a mathematical model to explain known facts and make new predictions. In this physics video lecture in hindi we talked about scalar field and vector field. For simplicity we will start with a scalar field instead of the vector. Notice that the divergence of a vector field is a scalar field. Field potentials and consideration of the helmholtz theorem for vector fields. Just as in scalar fields, the values of this vector function depend only on the points p in. In physics, scalar fields often describe the potential energy associated with a particular force.
698 1009 800 1241 992 464 370 457 738 1243 1177 1324 1074 466 122 810 238 1537 473 104 499 539 1526 420 1129 1178 1485 1304 1210 1494 652 727 1111 1147 1180 1286