Unit vectors in spherical polar coordinates x r sin. 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. The other courses of the program deal implicitly or explicitly with several coordinate systems and coordinate transformations, e. Let ul, u2 u3 represent the three coordinates in a general, curvilinear system, and let e. Sekhar chivukula for phy 415810 at michigan state university.
The conventional choice of coordinates is shown in fig. Velocity polar coordinates the instantaneous velocity is defined as. Polar coordinates lecture 24 integration and curvilinear. Two coordinate systems cylindrical polar coordinates and spherical polar coordinates will be illustrated. For instance, the point 0,1 in cartesian coordinates would be labeled as 1, p2 in polar coordinates. The name curvilinear coordinates, coined by the french mathematician lame, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. The particle is located by the radial distance r from a fixed point and by an angular measurement. Formulae for grad, div and curl in these coordinate systems are developed and the two most important examples, cylindrical and spherical polar coordinates are. Two coordinate systems cylindrical polar coordinates. The presentation here closely follows that in hildebrand 1976. 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. Advanced fluid dynamics 2017 navier stokes equation in. The gradient of v in spherical polar coordinates c.
These are two important examples of what are called curvilinear coordinates. Wellknown examples of curvilinear coordinate systems in threedimensional euclidean space r 3 are cylindrical and spherical polar coordinates. So polar coordinates are usually used for rotational motion in two dimensions. Unit vectors the unit vectors in the cylindrical coordinate system are functions of position. Curvilinear motion in polar coordinates it is sometimes convenient to express the planar twodimensional motion of a particle in terms of polar coordinates r. Basic trigonometry can be used to show that the cartesian and curvilinear comnponents are related as follows. F and f include the scale factors h u, h v and h w.
The main application is the subject of curvilinear coordinates. 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. These same examples are used to illustrate the general ideas. It is a simple matter of trigonometry to show that we can transform x,y. Orthogonal curvilinear coordinates and div, grad, curl. We will present polar coordinates in two dimensions and cylindrical and spherical coordinates in three dimensions. Lecture l5 other coordinate systems in this lecture, we will look at some other common systems of coordinates. If we express the position vector in polar coordinates, we get rt r rcos. Curvilinear coordinates spherical polar coordinates. We can equally introduce cylindrical polar coordinates which we will use here as the prime example for curvilinear coordinate systems. We learn about double and triple integrals, and line integrals and surface integrals. We can express the location of p in polar coordinates as r r u r.
Cylindrical coordinates transforms the forward and reverse coordinate transformations are. A point in coordinate space r is often represented as r x. Polar and curvilinear coordinates, kinematics due wednesday september 28, 2016 school of engineering. Cartesian coordinates we will start by studying the motion of a particle. Howard buck schreyer, who introduced me to curvilinear coordinates when i was a student in his continuum mechanics class back in 1987. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. The coordinates of a point p are given by r, the distance from p to o, and. We think of a particle as a body which has mass, but has negligible dimensions. R is the length of the vector, theta is the angle it makes with the xaxis. This section shows how to calculate these derivatives in other coordinate systems. A rigid body is rotating about a xed axis with a constant velocity take.
Chapter 2 introduces the most common coordinate systems and iterates the distinction between irregu lar bases and curvilinear coordinates. Here, to avoid all possible confusion with r in spherical polars, we will. 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. By using this reference surface, the coordinates of any. Many of the steps presented take subtle advantage of the orthogonal nature of these systems. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. Change of variables as we saw in lecture 16, in e2 we can use the polar coordinates system. For example in lecture 15 we met spherical polar and cylindrical polar coordinates.
The equation u j c j, where c j is a constant, represents a. Treating bodies as particles is, of course, an idealization which involves an approximation. Div, grad and curl in orthogonal 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. Cylindrical polar coordinates in cylindrical polar coordinates. The last is rather trivial but included to emphasize that this is a transformation in 3dimensional space. But in polar coordinates, we represent this vector in terms of r and theta. Note on curvilinear coordinates physics libretexts. Gradient, divergence and curl in curvilinear coordinates. Curvilinear coordinate an overview sciencedirect topics. The laplacian in curvilinear coordinates the full story.
Cartesian cylindrical spherical cylindrical coordinates x r cos. We shall see that these systems are particularly useful for certain classes of problems. For cartesian coordinates the normalized basis vectors are e. The laplacian in curvilinear coordinates the full story peter haggstrom. 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. Expression of strain tensor in orthogonal curvilinear coordinates 49 d surface nested in a cartesian coordinate system. Central force lecture 25 integration and curvilinear. The ranges of the variables are 0 coordinates can be written as 2.
If all motion components are directly expressible in terms of horizontal and vertical coordinates 1 also, dydx tan. Polar coordinates, parametric equations whitman college. 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. The common differential operators including the vector laplacian are expressed in arbitrary curvilinear coordinates. Expression of strain tensor in orthogonal curvilinear.
The relation between cartesian and cylindrical coordinates is given by. The sides of the small parallelepiped are given by the components of dr in equation 5. This video especially covers polar coordinates which is one method to analyze curvilinear motion. Vector and tensor algebra in threedimensional curvilinear coordinates note. Calculus with curvilinear coordinates web education. Me 230 kinematics and dynamics university of washington. Jul 10, 2012 an educational video in engineering mechanics. In order to do so it is convenient to start from the expression for the function di erential. 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. A system of curvilinear coordinates is used to parameterize the kinematic geometry of motion transmission between skew axes. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Construct the basis vectors for cylindrical coordinates, fe.
Navier stokes equation in curvilinear coordinate systems 1. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. In this section, we explore the question of how to quantize a system in curvilinear coordinates, using plane polar coordinates as an example. So far we have only used cartesian x,y,z coordinates.
Sometimes, because of the geometry of a given problem, it is easier to work in some other coordinate system. We have some vector, and we can represent it in x,y coordinates. Curvilinear analysis in a euclidean space mechanical engineering. Pdf tensor analysis and curvilinear coordinates researchgate. We would like to be able to compute slopes and areas for these curves using polar coordinates.
1021 36 119 813 1263 942 1138 1333 528 1241 1029 1329 1086 882 705 626 137 1322 1022 342 972 1291 431 202 828 1247 244 961 379 235 671 8 1075 265 964 922 821 803 135 763 1033 294 244