Div, grad, curl, and the laplacian the most common way that the gradient of a function, the divergence of a vector field, and the curl of a vector field are presented is entirely algebraic with barely any indication of what these mean. Formulae for grad, div and curl in these coordinate systems are developed and the two most important examples, cylindrical and spherical polar coordinates are. The name curvilinear coordinates, coined by the french mathematician lame, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Howard buck schreyer, who introduced me to curvilinear coordinates when i was a student in his continuum mechanics class back in 1987. We can equally introduce cylindrical polar coordinates which we will use here as the prime example for curvilinear coordinate systems. Vector and tensor algebra in threedimensional curvilinear coordinates note. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with cylindrical or spherical symmetry. Note that the radial direction, r, extends outward from the fixed origin, o, and the transverse coordinate, q,is measured counterclockwise ccw from the horizontal. This video especially covers polar coordinates which is one method to analyze curvilinear motion. Cylindrical polar coordinates in cylindrical polar coordinates. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Polar coordinates lecture 24 integration and curvilinear. We will present polar coordinates in two dimensions and cylindrical and spherical coordinates in three dimensions.
Dec 14, 20 this is video 23 on curvilinear motion in which i introduce the use and derive the equations of motion using a polar coordinate system. Div, grad and curl in orthogonal curvilinear coordinates. These are two important examples of what are called curvilinear coordinates. Up until now, a rectangular cartesian coordinate system has been used, and a set of orthogonal unit base vectors i e has been employed as. These same examples are used to illustrate the general ideas. Cartesian cylindrical spherical cylindrical coordinates x r cos.
Velocity polar coordinates the instantaneous velocity is defined as. Two coordinate systems cylindrical polar coordinates. In spherical coordinates, we specify a point vector by giving the radial coordinate r, the distance from the origin to the point, the polar angle, the angle the radial vector makes with respect to the zaxis, and the. Calculus with curvilinear coordinates web education. This section shows how to calculate these derivatives in other coordinate systems. If all motion components are directly expressible in terms of horizontal and vertical coordinates 1 also, dydx tan. The gradient of v in spherical polar coordinates c. Let ul, u2 u3 represent the three coordinates in a general, curvilinear system, and let e. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates.
Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. If we express the position vector in polar coordinates, we get rt r rcos. The laplacian in curvilinear coordinates the full story peter haggstrom. For instance, the point 0,1 in cartesian coordinates would be labeled as 1, p2 in polar coordinates. Polar and curvilinear coordinates, kinematics due wednesday september 28, 2016 school of engineering.
Wellknown examples of curvilinear coordinate systems in threedimensional euclidean space r 3 are cylindrical and spherical polar coordinates. We think of a particle as a body which has mass, but has negligible dimensions. The common differential operators including the vector laplacian are expressed in arbitrary curvilinear coordinates. Sekhar chivukula for phy 415810 at michigan state university. Polar coordinates, parametric equations whitman college. The relation between cartesian and cylindrical coordinates is given by. Sometimes, because of the geometry of a given problem, it is easier to work in some other coordinate system. Gradient, divergence and curl in curvilinear coordinates. The for the vector 3 i i x v r and tensor 35 ij i j ij k k xx s r xx r r. Expression of strain tensor in orthogonal curvilinear. Curvilinear analysis in a euclidean space mechanical engineering. Curvilinear coordinate an overview sciencedirect topics. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. Me 230 kinematics and dynamics university of washington.
A point in coordinate space r is often represented as r x. R is the length of the vector, theta is the angle it makes with the xaxis. Cylindrical coordinates transforms the forward and reverse coordinate transformations are. We have some vector, and we can represent it in x,y coordinates. Another reason to learn curvilinear coordinates even if you never explicitly apply the knowledge to any practical problems is that you will develop a far deeper understanding of cartesian tensor analysis. Navier stokes equation in curvilinear coordinate systems 1. The particle is located by the radial distance r from a fixed point and by an angular measurement. Unit vectors the unit vectors in the cylindrical coordinate system are functions of position. Unit vectors in spherical polar coordinates x r sin. Many of the steps presented take subtle advantage of the orthogonal nature of these systems. The last is rather trivial but included to emphasize that this is a transformation in 3dimensional space. So polar coordinates are usually used for rotational motion in two dimensions. Advanced fluid dynamics 2017 navier stokes equation in. Pdf tensor analysis and curvilinear coordinates researchgate.
In orthogonal curvilinear coordinates, the vector derivatives. So far we have only used cartesian x,y,z coordinates. A rigid body is rotating about a xed axis with a constant velocity take. For cartesian coordinates the normalized basis vectors are e. Orthogonal curvilinear coordinates and div, grad, curl. The coordinates of a point p are given by r, the distance from p to o, and. In this section, we explore the question of how to quantize a system in curvilinear coordinates, using plane polar coordinates as an example. Central force lecture 25 integration and curvilinear. A system of curvilinear coordinates is used to parameterize the kinematic geometry of motion transmission between skew axes.
For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. We can express the location of p in polar coordinates as r r u r. We learn about double and triple integrals, and line integrals and surface integrals. The sides of the small parallelepiped are given by the components of dr in equation 5. We would like to be able to compute slopes and areas for these curves using polar coordinates. Lecture l5 other coordinate systems in this lecture, we will look at some other common systems of coordinates. By using this reference surface, the coordinates of any. But in polar coordinates, we represent this vector in terms of r and theta. Note on curvilinear coordinates physics libretexts. The conventional choice of coordinates is shown in fig. Expression of strain tensor in orthogonal curvilinear coordinates 49 d surface nested in a cartesian coordinate system. Jul 10, 2012 an educational video in engineering mechanics.
Change of variables as we saw in lecture 16, in e2 we can use the polar coordinates system. Basic trigonometry can be used to show that the cartesian and curvilinear comnponents are related as follows. The laplacian in curvilinear coordinates the full story. The presentation here closely follows that in hildebrand 1976. Curvilinear motion in polar coordinates it is sometimes convenient to express the planar twodimensional motion of a particle in terms of polar coordinates r. Construct the basis vectors for cylindrical coordinates, fe. Cartesian coordinates we will start by studying the motion of a particle. F and f include the scale factors h u, h v and h w. Curvilinear coordinates spherical polar coordinates.
Vector v is decomposed into its u, v and wcomponents. The main application is the subject of curvilinear coordinates. Here, to avoid all possible confusion with r in spherical polars, we will. The other courses of the program deal implicitly or explicitly with several coordinate systems and coordinate transformations, e.
The cylindrical polar system is related to cartesian coordinates x. In order to do so it is convenient to start from the expression for the function di erential. In this video i have tried to explain the curvilinear coordinates i have explained spherical polar coordinate for any queries please comment in comment section. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. Treating bodies as particles is, of course, an idealization which involves an approximation. The equation u j c j, where c j is a constant, represents a. It is a simple matter of trigonometry to show that we can transform x,y. Chapter 2 introduces the most common coordinate systems and iterates the distinction between irregu lar bases and curvilinear coordinates.
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