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In spherical coordinates a point P is specified by. r where r is measured from the origin is, measured from the z axis and is measured from. the x axis or x z plane see figure at right With, z axis up is sometimes called the zenith angle. and the azimuth angle A vector at the point P, is specified in terms of three mutually. perpendicular components with unit vectors r, perpendicular to the sphere of radius r. perpendicular to the cone of angle and, perpendicular to the plane through the z axis at.

angle The unit vectors r form a right, handed set, Infinitesimal lengths and volumes. An infinitesimal length in the rectangular system is given by. dL d x2 d y2 d z2 1, and an infinitesimal volume by. dv dx dy dz 2, In the cylindrical system the corresponding quantities are. dL d r 2 r 2 d 2 d z 2 3, and d v d r r d d z 4, In the spherical system we have. dL d r 2 r 2 d 2 r 2 sin 2 d 2 5, and d v dr r d r sin d 6.

Direction cosines and coordinate system transformation. As shown in the figure on the right the, projection x of the scalar distance r on the x axis. is given by r cos where is the angle, between r and the x axis The projection of r on. the y axis is given by r cos and the, projection on the z axis by r cos Note that. so cos cos, The quantities cos cos and cos are, called the direction cosines From the theorem. of Pythagoras, cos 2 cos 2 cos 2 1 7, The scalar distance r of a spherical coordinate.

system transforms into rectangular coordinate, x r cos r sin cos 8. y r cos r sin sin 9, z r cos r cos 10, from which, cos sin cos 11. cos sin sin direction cosines 12, cos cos 13, As the converse of 8 9 and 10 the spherical coordinate values r may be. expressed in terms of rectangular coordinate distances as follows. r x2 y2 z2 r 0 14, cos 1 0 15, From these and similar coordinate transformations of spherical to rectangular and. rectangular to spherical coordinates we may express a vector A at some point P with. spherical components Ar A A as the rectangular components Ax Ay and Az where. Ax Ar sin cos A cos cos A sin 17, Ay Ar sin sin A cos sin A cos 18.

Az Ar cos A sin 19, Note that the direction cosines are simply the dot products of the spherical unit vector r. with the rectangular unit vectors x y and z, r x sin cos cos 20. r y sin sin cos 21, r z cos cos 22, These and other dot product combinations are listed in the following table. Rectangular Cylindrical Spherical, x y z r z r, x 1 0 0 cos sin 0 sin cos cos cos sin. Rectangular, y 0 1 0 sin cos 0 sin sin cos sin cos.

z 0 0 1 0 0 1 cos sin 0, r cos sin 0 1 0 0 sin cos 0. Cylindrical, sin cos 0 0 1 0 0 0 1, z 0 0 1 0 0 1 cos sin 0. r sin cos sin sin cos sin 0 cos 1 0 0, cos cos cos sin sin cos 0 sin 0 1 0. sin cos 0 0 1 0 0 0 1, Note that the unit vectors r in the cylindrical and spherical systems are not the same. For example, Spherical Cylindrical, r x sin cos r x cos.

r y sin sin r y sin, r z cos r z 0, In addition to rectangular cylindrical and spherical coordinate systems there are many. other systems such as the elliptical spheroidal both prolate and oblate and paraboloidal. systems Although the number of possible systems is infinite all of them can be treated. in terms of a generalized curvilinear coordinate system. The fundamental parameters of the rectangular cylindrical and spherical coordinate. systems are summarized in the following table, Coordinate Unit Length Coordinate. Coordinates Range, system vectors elements surfaces. x to x or i dx Plane x constant, Rectangular, y to y or j dy Plane y constant. z to z or k dz Plane z constant, r 0 to r dr Cylinder r constant.

Cylindrical 0 to 2 r d Plane constant, z to z dz Plane z constant. r 0 to r dr Sphere r constant, Spherical 0 to r d Cone constant. 0 to 2 r sin d Plane constant, The following two tables give the unit vector dot products in rectangular coordinates for. both rectangular cylindrical and rectangular spherical coordinates. x2 y2 x2 y2, x2 y2 x2 y2, Rectangular cylindrical product in rectangular coordinates. Example y cos, x2 y2 z 2 x2 y2 z 2 x2 y2 z 2, xz yz x2 y 2.

x2 y 2 x2 y 2 z 2 x2 y 2 x2 y 2 z 2 x2 y2 z 2, Rectangular spherical product in rectangular coordinates. Example x r sin cos, Here are the transformations of vector components between coordinate systems. Rectangular to cylindrical Cylindrical to rectangular. A Ax Ay Ay Ar sin A cos, Az Az Az Az, Rectangular to spherical. Ar Ax Ay Az, xz yz x2 y 2, A Ax Ay Az, x2 y 2 x2 y 2 z 2 x2 y 2 x2 y 2 z 2 x2 y 2 z 2. x2 y 2 x2 y 2, Spherical to rectangular, Ax Ar sin cos A cos cos A sin.

Ay Ar sin sin A cos sin A cos, Az Ar cos A sin, And here are expressions for the gradient divergence and curl in all three coordinate. Rectangular coordinates, Az Ay Ax Az Ay Ax, y z z x x y x y z. Cylindrical coordinates, 1 Az A Ar Az 1 Ar, r z z r r r r z. Spherical coordinates, A 2 r 2 Ar A sin, r r r sin r sin. 1 A 1 1 Ar 1 A, A r A sin rA rA r, r sin r sin r r r.

Review of Coordinate Systems A good understanding of coordinate systems can be very helpful in solving In the cylindrical system the corresponding quantities are