The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations, and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics. It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. It is usually denoted by the symbols ∇ ⋅ ∇ $\rho=$ constant is a sphere of radius $\rho$ centered at the origin.In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. Sphere of radius 3 centered at the origin. Most people don't have trouble understanding what $\rho=3$ means. Y &= \rho\sin\phi\sin\theta\label, we see it is only a half plane because $\rho\sin\phi$ cannot be negative. In summary, the formulas for Cartesian coordinates in terms of spherical coordinates are Since $r=\rho\sin\phi$, these components can be rewritten as $x=\rho\sin\phi\cos\theta$ and $y=\rho\sin\phi\sin\theta$. As $\theta$ is the angle this hypotenuse makes with the $x$-axis, the $x$- and $y$-components of the point $Q$ (which are the same as the $x$- and $y$-components of the point $P$) are given by $x=r\cos\theta$ and $y=r\sin\theta$. In the right plot, the distance from $Q$ to the origin, which is the length of hypotenuse of the right triangle, is labeled just as $r$. The cyan triangle, shown in both the original 3D coordinate system on the left and in the $xy$-plane on the right, is the right triangle whose vertices are the origin, the point $Q$, and its projection onto the $x$-axis. The distance of the point $Q$ from the origin is the same quantity. The length of the other leg of the right triangle is the distance from $P$ to the $z$-axis, which is $r=\rho\sin\phi$. As the length of the hypotenuse is $\rho$ and $\phi$ is the angle the hypotenuse makes with the $z$-axis leg of the right triangle, the $z$-coordinate of $P$ (i.e., the height of the triangle) is $z=\rho\cos\phi$. The pink triangle above is the right triangle whose vertices are the origin, the point $P$, and its projection onto the $z$-axis. We can calculate the relationship between the Cartesian coordinates $(x,y,z)$ of the point $P$ and its spherical coordinates $(\rho,\theta,\phi)$ using trigonometry. Lastly, $\phi$ is the angle between the positive $z$-axis and The angle between the positive $x$-axis and the line segment from the origin Point $Q$ is the projection of $P$ to the $xy$-plane, then $\theta$ is The coordinate $\rho$ is the distance from $P$ to the origin. Spherical coordinates are defined as indicated in theįollowing figure, which illustrates the spherical coordinates of the Relationship between spherical and Cartesian coordinates On this page, we derive the relationship between spherical and Cartesian coordinates, show an applet that allows you to explore the influence of each spherical coordinate, and illustrate simple spherical coordinate surfaces. The following graphics and interactive applets may help you understand sphericalĬoordinates better. But some people have trouble grasping what the If one is familiar with polar coordinates, then the angle $\theta$ isn't too difficult to understand as it is essentially the same as the angle $\theta$ from polar coordinates. Spherical coordinates determine the position of a point in three-dimensional space based on the distance $\rho$ from the origin and two angles $\theta$ and $\phi$. Spherical coordinates can be a little challenging to understand at first.
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |