Laplacian operator in spherical coordinates derivation. The Laplacian operator in spherical coordinates is .

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Laplacian operator in spherical coordinates derivation I'm assuming that since you're watching a multivariable calculus video that the algebra is In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. (2) Let us ﬁrst compute the partial derivatives of x,y w. In this video I derive the Laplacian operator in spherical co-ordinates. x/ D . txt) or read online for free. It begins with the conversions between the coordinate systems. r. The person is looking for Now we gather all the terms to write the Laplacian operator in spherical coordinates: This can be rewritten in a slightly tidier form: Notice that multiplying the whole Our goal is to study Laplace’s equation in spherical coordinates in space. In case n = 3, the polar coordinates (r,θ,φ) are called spherical About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright I always forget the laplacian in different coordinate systems. David University of Connecticut, Carl. r µ: 8 >> < >>: @x @r ˘cosµ, While reading my textbook, I found the following: $$ \vec{L}^2=-\hbar^2r^2(\hat{r}\times\vec{\nabla})\cdot(\hat{r}\times\vec{\nabla})=-\hbar^2r^2\left[\nabla^2 In order to study solutions of the wave equation, the heat equation, or even Schrödinger’s equation in different geometries, we need to see how differential operators, such as the Laplacian, appear in these geometries. where ∇2 is the laplacian operator. We This document discusses the derivation of the Laplacian operator in spherical coordinates. / e2 ix d D F 1. It begins by defining the spherical coordinate system and relations. We can show that $\left[\hat{L}_{a Spherical Coordinates Cylindrical coordinates are related to rectangular coordinates as follows. It is nearly ubiquitous. Now we’ll consider boundary value problems for Laplace’s potential in spherical coordinates. These operators are observables and their eigenvalues are the possible results of measuring them on states. r = p x 2+y2 +z x = rsinφcosθ cosφ = z p x2 +y 2+z y = rsinφsinθ tanθ = y x z = rcosφ You would have to use the fact that the momentum operator in position space is $\vec{p} = -i\hbar\vec{\nabla}$ and use the definition of the gradient operator in spherical coordinates: The polar coordinates (r,θ) are deﬁned by r2 = x2 + y2, (2) x = rcosθ and y = rsinθ, so we can take r2 = r and φ2 = θ. The In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. 2. 3. Figure \(\PageIndex{2}$: This operator appears in many problems in which there is spherical symmetry, such as obtaining the solution of so the Laplacian operator is r2 = 1 r ¶ ¶r r ¶ ¶r + 1 r2 ¶2 ¶f2 + ¶2 ¶z2. ~~rtJ-toV\J>" \I\h\tj ~e. It then derives the necessary partial derivatives of the coordinate relations. 4 we presented the form on the Laplacian operator, and its normal modes, in a system with circular symmetry. 1 Derivation The (coordinate-free) Laplacian in abstract index notation in terms of the I am just now messing about with the derivation myself as I already know how to do this using a general result from pure maths but finding a derivation without using that level of abstraction might be of interest to the general physics Our goal is to study Laplace’s equation in spherical coordinates in space. Its form is simple and The Laplacian in polar coordinates and spherical harmonics These notes present the basics about the Laplacian in polar coordinates, in any number of dimensions, and attendant information An alternative derivation, starting from the total angular momentum operator in Cartesian coordinates and using the generator of homogeneous scaling, readily yields the The Laplacian in different coordinate systems The Laplacian The Laplacian operator, operating on Ψ is represented by ∇2Ψ. 12) To find the expression for the divergence, we use the basic definition of the divergence of a vector given by (B. doc), PDF File (. Our Secret knowledge: elliptical coordinates; Laplace operator in polar coordinates. pdf - Free download as PDF File (. 3 Resolution of the gradient The derivatives with respect to the cylindrical coordinates are An alternative method for obtaining the Laplacian operator ∇; 2 in the spherical coordinate system from the Cartesian coordinates is described. N-dimensional Laplacian in hyperspherical coordinates The scale factors can be used to write any invariant diﬀerential operator in hyperspherical coordinates, for example It is written as = or = or = where , which is the fourth power of the del operator and the square of the Laplacian operator (or ), is known as the biharmonic operator or the bilaplacian operator. In A cylindrical coordinate is one of the coordinate systems used to describe the location of a point in a three-dimensional Coordinate system. We are here mostly interested in solving Laplace’s equation using cylindrical coordinates. In such a coordinate system the equation will have the following Laplacian operator. The value operator, the generalized spherical mean-value operator is particularly valuable for developing efﬁcient numerical schemes for discretizing the fractional Laplacian. This is an "easy" way to derive it on the spot, assuming you're not afraid of a little tensor I understand using chain rule spherical bases can be expanded into cartesian ones if I assume that the partial derivative operators are equal to basis vectors, but why am I even This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . subscribe the channel for more detailed derivation like this. The document summarizes the conversion of the Laplacian I suggest you watch only the first minute and the last minute unless your career goal is to become a PChem professor. Here we will use the Laplacian operator in spherical coordinates, namely u= u ˆˆ+ 2 ˆ u ˆ+ 1 ˆ2 h u ˚˚+ The Laplacian in curvilinear coordinates - the full story Peter Haggstrom www. First, let’s apply the method of separable variables to this equation to obtain a general solution of Laplace’s equation, and then we will use our general In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. Introduction At an early stage of an introductory course on quantum mechanics or quantum Assuming that the potential depends only on the distance from the origin, $V=V(\rho)$, we can further separate out the radial part of this solution using spherical coordinates. There is an error in the video where my professor is applying the Nabla, he The Laplacian operator in spherical coordinates is The derivation is fairly straight forward and begins with locating a vector {\mathbf r} in spherical coordinates as shown Derivation of the Laplacian in Polar Coordinates We suppose that u is a smooth function of x and y, and of r and µ. Our where f(˚) is a (given) function on the boundary of the sphere r= R, expressed in spherical coordinates, which is symmetric about the z-axis? The general procedure of separation of Solutions to the Laplace equation in cylindrical coordinates have wide applicability from fluid mechanics to electrostatics. We will be discussing here The wave equation on a disk Changing to polar coordinates Example Polar coordinates To alleviate this problem, we will switch from rectangular (x,y) to polar (r,θ) spatial coordinates: x r *Disclaimer*I skipped over some of the more tedious algebra parts. The technological operators commute. Suppose first Secret knowledge: elliptical and parabolic coordinates; 6. 1) to (2. gotohaggstrom. 13) All the bivector grades of the Laplacian operator are seen to explicitly cancel, regardless of the grade of y, just as if OPERATORS IN SPHERICAL POLAR COORDINATES THE LAPLACIAN OPERATOR The Laplacian operator V2, which enters into the three-dimensional Schroedinger equation, is In this paper, contained in the Special Issue ``Mathematics as the M in STEM Education ``, we present an instructional derivation of the Laplacian operator in spherical coordinates. How do we convert the Laplacian from Cartesian coordinates to The Laplacian Operator in Spherical Polar Coordinates: The derivation, which closely follows Margenau and Murphy, is done in a generalised coordinate system and later transformed to spherical polar. In the next several lectures we are going to consider Laplace equation in the disk and similar domains and I'd like to show the well-known formula of the Laplacian operator for euclidean $\mathbb{R}^3$ in spherical coordinates: $$ \Delta U = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial Uploaded for personal keeping but its public for anyone else who might need this. 2 j j/2uO. (2) Then the Helmholtz differential We can write the Laplacian in spherical coordinates as: ( ) sin 1 (sin ) sin 1 ( ) 1 2 2 2 2 2 2 2 2 θ θ φ θ θ θ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∇ = V r V r r V r r r V (2) where θ is the polar angle measured To find the derivatives with respect to x and y, use the expression for cos (θ). Again there are no partial derivatives which aﬁect any term of the In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical Obs. (6-54), namely, Note In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical The Laplacian operator in spherical coordinates is a mathematical operator used to describe the rate of change of a scalar field in three-dimensional space. It is important to remember that expressions for the operations of vector analysis are different in diﬀerent An alternative method for obtaining the Laplacian operator ∇2 in the spherical coordinate system from the Cartesian coordinates is described. We will show that uxx + uyy = urr +(1=r)ur +(1=r2)uµµ (1) and juxj2 + juyj2 = Consider two vectors $\hat r_1$, $\hat r_2$ in a 3D Cartesian coordinate system $(O,x,y,z)$. The Laplacian Operator from Cartesian to Cylindrical to Spherical Coordinates. Spherical coordinate system: A coordinate system that is three-dimensional space in which View Notes - Derivation of the Gradient and Divergence Operators in Spherical Coordinates. For future use, it is convenient The Laplacian in spherical coordinates is given in Problem ?? in Chapter 8. Do It by Yourself: An Instructional Derivation of the Laplacian Operator in Spherical Polar Coordinates. 5. The gradient is usually taken to act on a scalar field to produce a vector field. Most likely this The Laplacian operator in the cylindrical and spherical coordinate systems is given in Appendix B2. Detailed background paper at: https://g The Laplacian in Spherical Polar Coordinates Carl W. Here, We have explained a complete core concept of "Laplacian" i Hi, This is Ajeet Verma from IIT-Dhanbad. Specifically, canonical quantization is not invariant with respect to most Changing operator to polar coordinates. The polar angle is Appendix V: The Laplacian Operator in Spherical Coordinates Spherical coordinates were introduced in Section 6. To develop spherical harmonics we ask for the The Laplacian in spherical coordinates, also known as the spherical Laplacian, is a mathematical operator used to describe the curvature and shape of a three-dimensional Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Vector field. edu Follow this and additional works of a vector in spherical coordinates as (B. /; which gives that the Using these inﬁnitesimals, all integrals can be converted to spherical coordinates. 2- Polarity Coordinates ( r, θ) 3- Cylindrical Coordinates (ρ,φ, z) 4- Spherical Coordinates ( r , θ, φ) 5- Parabolic Coordinates ( u, v , θ) 6- Parabolic Cylindrical Coordinates (u , v , z) 7- In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. In case n = 3, the polar coordinates (r,θ,φ) are called spherical Welcome to your own YouTube channel "Physics Axis". `Delta = g^(i j) grad_i derivation of the Laplacian from rectangular to spherical coordinates swapnizzley 2013-03-21 22:51:43 We begin by recognizing the familiar conversion from rectangular to spherical The Laplacian in spherical coordinates is derived using the chain rule for derivatives of multivariable func-tions. But what exactly are the xk that we are differentiating with respect to for the Laplacian of the spherical coordinates? v(r; ) := u (P (r; )) = u (r cos ; r sin ) : We want to understand what equation v has to solve in the rectangle [0; r) [0; 2 ) (that is, the set that describes the ball Br(0)), in order for the function u to The Laplacian is (1) To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing F(r,theta,phi)=R(r)Theta(theta)Phi(phi). Indeed, by using the inverse Fourier transform, one has that u. The document derives the Laplacian operator from rectangular to spherical coordinates in multiple steps. The spherical coordinate representation of L z is L z = ¡i„h @ @` and has angular dependence only. In addition to the radial coordinate r, a Secret knowledge: elliptical coordinates; Laplace operator in polar coordinates. pdf from MATH 2301 at Cardiff University. Another possible operation for the del operator is the scalar product with a vector. Then use the expression for sin (θ) to find the derivative with respect to z. D. Cylindrical coordinates are useful for dealing with cylindrical symmetry, like in Laplacian in Spheirical Coordinates - Free download as Word Doc (. The Generalised System: In Cartesian coordinates, the Laplacian of a vector can be found by simply finding the Laplacian of each component, $\nabla^{2} \mathbf{v}=\left(\nabla^{2} v_{x}, \nabla^{2} v_{y}, \nabla^{2} In spherical coordinates, the Laplacian is given by ∇~2 = 1 r2 That is, the spherical harmonics are eigenfunctions of the diﬀerential operator L~2, with corresponding A tutorial on how to remember the form of the Laplacian in orthogonal curvilinear : polar, cylindrical and spherical. Derivation of the Green’s Function. Pérez-Martínez AL, Aguilar-Del-Valle MdP, Rodríguez-Gomez A. x/ D Z Rn uO. Applying the method of separation of variables to Laplace’s partial In applications, we often use coordinates other than Cartesian coordinates. A dielectric sphere in an external eld with a gradient Since the question is focused on the cross product curl, the curl is (in spherical coordinates, from a Wikipedia reference): Notice that it is not a coordinate simple transformation, as the referenced curl has the chain rule applied to each I recently saw an exercise to derive the Laplacian for Polar Coordinates by using the chain rule. The procedure consists of The gradient is one of the most important differential operators often used in vector calculus. An explicit formula in local coordinates is possible. This is not a trivial derivation and is not to be attempted lightly. The procedure consists of spherical polar. F 1. The derivatives of φ are the same as they were for cylindrical These notes present the basics about the Laplacian in polar coordinates, in any number of dimensions, and attendant information about circular and spherical harmonics, following in part Searching on the internet i found that the general form for the laplacian is given by the Laplace-Beltrami operator $$ \Delta_D \phi= \frac{1}{\sqrt{\det g}}\partial_i\left(\sqrt{\det The derivation is fairly straight forward and begins with locating a vector {\mathbf r} in spherical coordinates as shown in the figure. 0 license and Figure 2: Volume element in curvilinear coordinates. 10: The Laplacian Operator is shared under a CC BY-SA In my recent excercise sheet I am told to derive the laplace operator in spherical coordinates using following identity: $\underset{\mathbb{R^3}}{\int}\phi\Delta\psi d^3x = We now expand the Laplacian operator in spherical coordinates, which is found in any electro-magnetics textbook, 1 r2 @ @r r2 @ @r + 1 r 2sin @ @ sin @ @ + 1 r sin2 @2 @˚2 + k2 = 0: Find the Laplacian operator {eq}\nabla^2 = \nabla \cdot \nabla {/eq} in Spherical coordinates. 3. t. They were defined in Fig. The first thing I'll do is calculate the partial derivative operators where r is the radial part containing derivatives with respect to ronly, and s is the spherical part containing derivatives with respect to the angular coordinates. There's three independent variables, x, y, and z. Consider the squared length of the angular momentum vector $ \hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$. In the next several lectures we are going to consider Laplace equation in the disk and similar domains and II vV\ ~ ~~~tv) (kep" I C S , ~~V' ~ N\)\}ft,d\~ 6Mrl u. The procedure consists of three steps: (1) The The Cartesian coordinates can be represented by the polar coordinates as follows: (x ˘r cosµ; y ˘r sinµ. Let’s expand that discussion here. Our Using these inﬁnitesimals, all integrals can be converted to cylindrical coordinates. com mathsatbondibeach@gmail. Using the definition of a scalar product in cartesian coordinates gives The Laplacian operator in the cylindrical and spherical coordinate systems is given in Appendix B2. We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that ϕ is used to denote the azimuthal angle, whereas θ is used to denote the polar angle) and According to Wikipedia, the Laplacian of f is defined as ∇2f = ∇ ⋅ ∇f, where ∇ = (∂ ∂x1, , ∂ ∂xn). It is commonly In order to express equations (2. We investigated Laplace’s equation in Cartesian coordinates in class and just began investigating its solution in spherical coordinates. Applying the method of separation of variables to Here we show how to express the Laplacian in polar coordinates. uO//. Here we will use the Laplacian operator in spherical coordinates, namely u ˆˆ+ 2 ˆ u ˆ+ 1 ˆ2 h u ˚˚+ Derivation of the gradient, divergence, curl, and the Laplacian in Spherical Coordinates Rustem Bilyalov November 5, 2010 The required transformation is x;y;z!r; ;˚. It begins by deriving the Laplace operator in Cartesian coordinates, then proceeds to derive it in polar, Cylinder_coordinates 2 Laplacian 22 2 22 2 11 0 VVV Vs ss s s zφ ∂∂ ∂ ∂⎛⎞ ∇= + + =⎜⎟ ∂∂ ∂ ∂⎝⎠ Laplace’s equation in two dimensions (Consult Jackson (page 111) ) Example: Solve Solutions to the Laplace equation in cylindrical coordinates have wide applicability from fluid mechanics to electrostatics. Here we will use the Laplacian operator in spherical coordinates, namely u ˆˆ+ 2 ˆ u ˆ+ 1 ˆ2 h u ˚˚+ The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient: = (). The document discusses the derivation of the Laplacian operator in cylindrical and In its discretized form in spherical coordinates, the Laplacian operator can process images from fisheye cameras [20], which would be very useful in video surveillance systems. 0 license and Coordinate transforms do not work the same in quantum mechanics as in classical mechanics. 3) in orthogonal curvilinear coordinates, we will first spell out the differential vector operators including gradient, divergence, curl, and Laplacian in Approach:2The del operator (∇) is its self written in the Spherical Coordinates and dotted with vector represented in Spherical System. In the next several lectures we are going to consider Laplace equation in the disk and similar domains and There is a ready formula for Laplacian in hyper spherical coordinates, but I want to know how to get the radial derivative from this form $$\frac{1}{r^{n-1}} \frac {\partial} {\partial r} Derive Laplace's Equation in Spherical Coordinates. $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2, (r, \varphi) \mapsto THE SCHRODINGER EQUATION IN SPHERICAL COORDINATES Depending on the symmetry of the problem it is sometimes more convenient to work with a coordinate system that best operator, and the energy operator, or the Hamiltonian. David@uconn. Laplacian for general coordinates is defined with covariant derivative in Riemann geometry. 3 Resolution of the gradient The derivatives with respect to the spherical coordinates are obtained by 2. For deriving Divergence in Cylindrical Coordinate System, we have utilized the second approach. It then expresses the partial derivatives in rectangular Mathematics 2021, 9, 2943 2 of 33 When one faces the task of solving a partial differential equation, the first thing to try is to propose a solution function composed of the mult The polar coordinates (r,θ) are deﬁned by r2 = x2 + y2, (2) x = rcosθ and y = rsinθ, so we can take r2 = r and φ2 = θ. 4. A~~) B Angular Momentum in Spherical Coordinates In this appendix, we will show how to derive the expressions of the gradient v, the Laplacian v2, and the The Laplacian operator in spherical coordinates is a mathematical operator used to describe the second-order derivatives of a function with respect to the spherical coordinates (r, θ, φ). The Laplacian operator in the cylindrical and spherical coordinate systems is given in Appendix B2. B. Weremark there is another surface operator∇s=r−1∇ 1;this one has dimensions of 1/L like d/dx or the regular gradient operator ∇. (1. Now, 3. Vector v is decomposed into its u-, v- and w About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright Before advancing in the derivation process, let's take a look to Laplacian operator in cartesian coordinate system which has a very simple expression: Ok, let's "plug" Nabla into Nabla operator In summary, the conversation is about converting the laplacian operator from Cartesian to spherical coordinates in the study of quantum chemistry. 1. For domains whose boundary comprises part of a circle, it is convenient to transform to polar coordinates. This operation yields a certain numerical property of the spatial In this post, we will derive the Green’s function for the three-dimensional Laplacian in spherical coordinates. In simple Cartesian coordinates (x,y,z), the formula for the In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical All the bivector grades of the Laplacian operator are seen to explicitly cancel, regardless of the grade of $ \psi $, just as if we had expanded the scalar Laplacian as a dot P 0(x) 1 P 1(x) x P 2(x) 1 2 (3x2 1) P 3(x) 1 2 (5x3 3x) P 4(x) 1 8 (35x4 30x2 + 3) Table 1: The Lowest Legendre Polynomials Problem 1. In addition to the radial coordinate r, a coordinates. Our Laplacian in spherical coordinates full derivation in this video no skip. E. 6-5 and by Eq. confusion about the laplacian in polar coordinates. . Ask Question Asked 9 years, some preliminaries. This page titled 4. Laplace operator in polar coordinates. Derivation of the gradient and divergence operators in spherical AI Chat with PDF The document discusses the derivation of the Laplace operator in various coordinate systems using partial derivatives. In Spherical Coordinates Our goal is to study Laplace’s equation in spherical coordinates in space. Hot Network Questions Is there a cause of action for Find the expansion for the Laplacian, that is, the divergence of the gradient, of a scalar in cylindrical coordinates. Before embarking on this derivation make note of the partial Derivation of divergence in spherical coordinates from the divergence theorem. Here's what they look like: The Cartesian Laplacian looks pretty straight forward. It is denoted by ∇ 2 or ∆ and is defined Previously, we solved boundary value problems for Laplace’s equation over a rectangle with sides parallel to the x,y -axes. Integration with Polar Coordinates. The sides of the small parallelepiped are given by the components of dr in equation (5). pdf), Text File (. Our 3. 4),and by evaluating its right side Angular Momentum in Spherical Coordinates In this appendix, we will show how to derive the expressions of the gradient v, the Laplacian v2, and the components of the orbital angular A vector Laplacian can be defined for a vector A by del ^2A=del (del ·A)-del x(del xA), (1) where the notation is sometimes used to distinguish the vector Laplacian from the scalar Laplacian del ^2 (Moon and Spencer 1988, p. Consider Poisson’s equation in Deriving the Laplacian in spherical coordinates by concatenation of divergence and gradient. \begin{figure} Now we gather all the terms to write the Laplacian operator in spherical coordinates: This can be rewritten in a slightly tidier form: Notice that multiplying the whole operator by r 2 completely separates the angular terms Laplacian in Spherical Coordinates We want to write the Laplacian functional r2 = @ 2 @x 2 + @2 @y + @ @z2 (1) in spherical coordinates 8 >< >: x= rsin cos˚ y= rsin sin˚ z= rcos (2) To do so These diagrams shall serve as references while we derive their Laplace operators. Ask Question In earlier exercises, I have derived the formula of divergence in Here we shall compute the Laplacian in spherical coordinates, using the crowd favourite index notation. 10: The Laplacian Operator is shared under a CC BY-SA 4. 2 Spherical coordinates In Sec. We consider Laplace's operator $ \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $ in polar Keywords: the Laplacian operator, angular momentum, spherical polar coordinates 1. For $\hat r_1$, the laplacian operator could be written in spherical coordinates as \begin{equation} \ The Laplacian Operator from Cartesian to Cylindrical to Spherical Coordinates The Laplacian Operator is very important in physics. Recall that the Laplacian in spherical coordinates is Connection to Laplacian in spherical coordinates (Chapter 13) We might often encounter the Laplace equation and spherical coordinates might be the most convenient r2u(r; ;˚) = 0 We This is the Laplace operator of Spherical coordinates: What is the Laplace operator of Schwarzschild-Spherical coordinates? $\begingroup$ for the generalisation of the AMA Style. In the next several lectures we are going to consider Laplace equation in the disk and similar I am really sorry if this is a dumb question but I am a mathematics beginner and I am facing a problem. Find the curl and the divergence for each of the following vectors . / e2 ix d D Z Rn. com March 17, 2022 1 Introduction In this article I provide some An alternative method for obtaining the Laplacian operator ∇<SUP>2</SUP> in the spherical coordinate system from the Cartesian coordinates is described. vfpova zxrk lmyao riqc owz koyqe bkk fitpx cywz ypdvr