A1:Orthogonal Coordinate Systems
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1 A1:Orthogonal Coordinate Systems A1.1 General Change of Variables Suppose that we express x and y as a function of two other variables u and by the equations We say that these equations are defining a transformation (or mapping) from points (u,v) in a uv-cartesian plane to points (x,y) in the xy-plane. The transformation is one-to-one from the set S in the uv-plane onto the set D in the xy-plane provided that: (i) Every point in S gets mapped to a point in D (ii) Every point in D is the image of a point in S, and (iii) Different points in S get mapped to different point in D With these restrictions, the defining equations can be solved for u and v as function of x and y by the inverse transformation as The Implicit Function Theorem guarantees that the system of equations can be solved for certain variables as function of other variables under the circumstance that the Jacobian determinant (J) satisfies A one-to-one transformation can be used to transform a double integral of the form to a double integral over the corresponding set S in the uv-plane. Under the transformation, the integrand f (x,y) becomes g(u,v) = f (x(u,v),y(u,v)). Now we have to discover how express the area element da = dxdy in terms of the area element dudv in the uv-plane. For any fixed value of u, the transformation equations define a parametric curve (with v as variable) in the xy-plane called a u-curve. Similarly, for a fixed v the equations define a parametric curve called a v-curve. Consider the differential area element bounded by the u-curves corresponding to nearby values u and u + du and the v-curves with nearby values v and v + dv. The image in the xy-plane of the area element dudv in the uv-plane is shown in Figure A.1
2 The vector PQ along the v-curve is given by Figure A.1: Differential area element Applying the Chain Rule we have that a change in x and y can be written as However, dv = 0 along the v-curve and PQ reduces to Similarly, the vector PR can be expressed as Taking the limit as du and dv approach zero the area element is approximately a parallelogram with an area given by Note that the above expression involves the absolute value of the Jacobian determinant (J) defined earlier. Consequently, if the functions x and y have continuous first partial derivatives with respect to u and w on the domain S, under the given transformation the double integral over D can be written as a double integral over S, hence
3 The change of variables extends from a double integral to triple (and higher order) integrals. Consider the transformation Near any point where the Jacobian (J) is nonzero, the transformation scales volume elements according to Under the assumption that x, y and z have continuous first partial derivatives with respect to u, v and w and that the transformation is one-to-one, then Note that it is not necessary for S or D to be closed or that the transformation is one-to-one at the boundary of S. A1.2 Orthogonal Curvilinear Coordinates We denote by xyz-space the system of Cartesian coordinates (x,y,z) in R 3. A different system of coordinates [u,v,w] in xyz-space can be defined by a continuous transformation of the form If the transformation is one-to-one from a region D in uvw-space onto a region R in xyz-space, then a point P in R can be represented by a triple [u,v,w], a unique point Q from the corresponding Cartesian uvw-space that the transformation maps to P. In this case we say that the transformation defines a curvilinear coordinate system in R and call [u,v,w] the curvilinear coordinates of P with respect to that system. Note that [u,v,w] are Cartesian coordinates in their own space (uvw-space), but are curvilinear coordinates in xyz-space. Typically it is reasonable to require the transformation to be only locally one-to-one. Thus, there may be more than one point Q in some small sub region of D that gets mapped to a point P by the transformation. Points that are not even locally one-to-one are called singular points. Now let [u,v,w] be a curvilinear coordinate system in xyz-space, and let P 0 be a nonsingular point for the system. The transformation is locally one-to-one near P 0. Let P 0 have curvilinear coordinates [u 0,v 0,w 0 ]. The plane with equation u = u 0 in uvw-space gets mapped by the transformation to a surface in xyz-space passing through P 0. We call this surface a u-surface with the parametric equations
4 Similarly, we have the v-surface v = v 0 and w-surface w = w 0. We say that [u,v,w] is an orthogonal curvilinear coordinate system in xyz-space if, for every nonsingular point P 0 in xyz-space, each of three coordinate surfaces u = u 0, v = v 0, and w = w 0 intersect the other two at P 0 at right angles. Pairs of coordinate surfaces through a point intersect along a coordinate curve through that point. The coordinate surfaces v = v 0 and w = w 0 intersect along the u-curve with parametric equations A unit vector u tangent to the curve u-curve through P 0 is normal to the coordinate surface u = u 0. Similar statements hold for the unit vectors v and w. For an orthogonal curvilinear coordinate system, the three vectors u, v, and w form a local basis of mutually perpendicular unit vectors at any non singular point P 0. A1.2.1 Scale Factors and Differential Elements Assume that [u,v,w] are orthogonal curvilinear coordinates in the xyz-space, the coordinate surfaces are smooth at any nonsingular point, and that the local basis vectors u, v, and w at any such point from a right-handed-triad. The position vector of a point P in xyz-space can be expressed in terms of the curvilinear coordinates as If we hold v = v 0 and w = w 0 fixed and let u vary, then r = r (u,v 0,w 0 ) defines a u-curve in xyzspace. At any point P on this curve, the vector is tangent to the u-curve at P. In general, the three vectors are tangent respectively to the, u-curve, v-curve, and the w-curve through P. They are also normal respectively to the u-surface, v-surface, and w-surface, so they are mutually perpendicular. The lengths of these three vectors are called the scale factors of the coordinate system. The scale factors are nonzero at a nonsingular point P, so the local basis at P can be obtained by dividing the tangent vectors of the coordinate curves by their lengths.
5 Hence The volume element in a orthogonal curvilinear coordinate system is the volume of an infinitesimal coordinate box bounded by pairs of u-, v-, and w-surfaces corresponding to values u and u + du, v and v + dv, and w and w + dw. The coordinate box is spanned by the vectors Therefore, the volume element is given by Furthermore, the surface area elements on coordinate surfaces are the respective faces of the coordinate box The arc length elements along the u-, v-, and w-curves are the edges of the coordinate box
6 A1.3 Polar Coordinates [r,θ] A point P with Cartesian coordinates (x,y) can be located by its polar coordinates [r,θ] in R² via the transformation where r is the distance from origin O to the point P and θ is the angle OP makes with the positive x-axis. Positive angles are measured counterclockwise in the interval [0,2π], see Figure A.2. Figure A2: Cartesian- to polar-coordinates The transformation is locally one-to-one from D, the half of the rθ-plane where 0 < r <, to the region R consists of all points in the xy-plane except from the origin. The origin can namely be represent by [0,θ] for any angle θ. Since the transformation is not locally one-to-one at r = 0, we regard the origin in the xy-plane as an singular point for the polar coordinate system. For any nonsingular fixed point [r 0,θ 0 ], the coordinates curves intersect each other at right angles. As expected, [r,θ] is an orthogonal curvilinear coordinates system in the xy-pane. The position vector r = r(r,θ) of a point P in the xy-plane can be expressed in terms of polar coordinates as At any point P, the vectors are tangent to the r- and θ-curve respectively at P. Thus, the local basis consists of the vectors
7 The area element can be determined as and the arc length elements A1.4 Cylindrical Coordinates [r,θ,z] The cylindrical coordinate system [r,θ,z] in R 3 is defined by the transformation where ordinary plane polar coordinates are used in the xy-plane while retaining the Cartesian z coordinate for measuring vertical distance. Figure A3a shows a point P located by its cylindrical coordinates [r,θ,z] and by its Cartesian coordinates (x,y,z). a) b) Figure A3: a) Cylindrical coordinates of a point b) Coordinate surfaces This transformation maps the half space D given by r > 0 onto all of xyz-space excluding the z- axis, and is locally one-to-one. All points on the z-axis are singular points of the system since the points [0,θ,z] are identical for any θ. The coordinate surfaces, shown in Figure A3b, are r-surfaces: circular cylinders with axis along the z-axis θ-surfaces: vertical half-plane radiating from the z-axis z-surfaces: horizontal planes
8 which for every nonsingular point intersect each other at right angles and thereby form an orthogonal curvilinear coordinate system. The coordinate curves are: r-curves: horizontal straight half-lines radiating from the z-axis θ-curves: horizontal circles with center on the z-axis z-curves: vertical straight lines The position vector of a point P in the xyz-plane can be expressed in terms of cylindrical coordinates as from which we can form the (right handed) local basis vectors The volume element dv, the area elements on coordinate surfaces, and the arc length elements on coordinate curves are A1.5 Spherical coordinates [ρ,ϕ,θ] In the system of spherical coordinates a point P in xyz-space is represented by the ordered triple [ρ,ϕ,θ] via the transformation It is conventional to consider spherical coordinates restricted in such a way that ρ > 0, 0 ϕ π, and 0 θ 2π (or -π < θ π). Every point not on the z-axis has exactly one spherical representation. Thus, all points on the z-axis are considered as singular points. Figure A4 shows a point P located by its cylindrical coordinates [ρ,ϕ,θ] and by its Cartesian coordinates (x,y,z). Note that the r coordinate in cylindrical coordinates is related to ρ and ϕ by
9 Figure A4: Spherical coordinates of a point For the spherical coordinate system, the coordinate surfaces are: The coordinate curves are: (i) ρ-surfaces: spheres centered at the origin (ii) ϕ-surfaces: vertical cones with vertices at origin (iii) θ-surfaces: vertical half-planes radiating from the z-axis (i) ρ-curves: half-lines radiating from the origin (ii) ϕ-curves: vertical semicircles with centers at the origin (iii) θ-curves: horizontal circles with centers on the z-axis Indeed, for every nonsingular point, the coordinate surfaces intersect each other at right angles. The position vector of a point P in the xyz-plane can be expressed in terms of spherical coordinates as from which we can form the (right handed) local basis vectors The volume element dv, the area elements on coordinate surfaces, and the arc length elements on coordinate curves are
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