![]() This article will use the ISO convention frequently encountered in physics: ( r, θ, φ ). The use of symbols and the order of the coordinates differs among sources and disciplines. The polar angle is often replaced by the elevation angle measured from the reference plane towards the positive Z axis the depression angle is the negative of the elevation angle. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle. The radial distance is also called the radius or radial coordinate. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin its polar angle measured from a fixed polar axis or zenith direction and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane. In this image, r equals 4/6, θ equals 90°, and φ equals 30°. A globe showing the radial distance, polar angle and azimuthal angle of a point P with respect to a unit sphere, in the mathematics convention. As in physics, ρ ( rho) is often used instead of r, to avoid confusion with the value r in cylindrical and 2D polar coordinates. The meanings of θ and φ have been swapped compared to the physics convention. Spherical coordinates ( r, θ, φ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ. This is the convention followed in this article. $h_r=1, h_\theta=r, h_\phi=r \sin \theta $ so $J(r,\theta,\phi)=r^2\sin\theta$įor cylindrical $h_\rho=1,h_\phi=\rho,h_z=1$ etc.3-dimensional coordinate system Spherical coordinates ( r, θ, φ) as commonly used in physics ( ISO 80000-2:2019 convention): radial distance r ( slant distance to origin), polar angle θ ( theta) (angle with respect to positive polar axis), and azimuthal angle φ ( phi) (angle of rotation from the initial meridian plane). For spherical coordinate system it is easy to show that: For an orthogonal system the jacobian is the product of scale factors. The factor $r$ which converts the infinitesimal change in $\theta$ ti the corresponding displacement $rd\theta$ is called the scale factor $h_\theta$. When you take the limits, this can be approximated as a rectangle, so, area is $\Delta r (r \Delta \theta)$. Now draw another constant curve $r+\Delta r$ and $\theta + \Delta \theta$. This is a ray which goes from the origin to infinity making an angle $\theta$ with the positive x axis. To locate a curve start with the constant coordinate $r$ which is a circle of radius $r$. ![]() Finally take the limit in $\Delta x ,\Delta y\rightarrow 0$ and this is understood when you write $\int \int dx dy$. Your area element would be a region bounded by the lines $x,x+\Delta x,y, y+\Delta y$ and your area element is $\Delta x \Delta y$. ![]() Please note, that all this is a part of nonstandard analysis which exploits the notion of an infinitesimal. ![]() The Jacobian is just a product of the scale factors when you change from one (orthogonal) coordinate system to another. Jacobians are used extensively in Statistical mechanics, but usually most physics problems come in flavors of one in three coordinate systems(cart, spherical, cylindrical) so we do not bother with the Jacobian when geometric intuition gets us the right answer for these three cases readily, which I have written in the comment. So in general your area element is $dA=\det du dv$ $$\iint_S f(x,y) dx dy = \iint_S f(x(u,v),y(u,v)) \det du dv$$įor 2-D polar coordinates $u=r$ and $v=\theta$ ![]()
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