$\endgroup$ – Cameron Williams Apr 28 '15 at 14:02 ∇f = ∂2 xf +∂ 2 yf +∂ 2 zf, (1) where the last expression is given in Cartesian coordinates, and ∂2 xf means ∂2f/∂x2, etc. The textbook shows the form in cylindrical and spherical coordinates. $\begingroup$ The Laplacian takes a scalar valued function and gives back a scalar valued function. The Laplacian is a good scalar operator (i.e., it is coordinate independent) because it is formed from a combination of divergence (another good scalar operator) and gradient (a good vector operator). The Laplacian operator for a Scalar function is defined by (1) in Vector notation, where the are the Scale Factors of the coordinate system. The divergence of the gradient of a scalar function is called the Laplacian. The Laplacian in Spherical Polar Coordinates Carl W. David University of Connecticut, Carl.David@uconn.edu This Article is brought to you for free and open access by the Department of Chemistry at DigitalCommons@UConn. The LaPlacian. The Scalar Laplacian The scalar Laplacian is simply the divergence of the gradient of a scalar field: ∇⋅∇g(r) The scalar Laplacian therefore both operates on a scalar field and results in a scalar field. In Tensor notation, the Laplacian is written (2) where is a Covariant Derivative and (3) The finite difference form is (4) LaPlacian in other coordinate systems Often, the Laplacian is denoted as “∇2”, i.e. : ∇∇⋅∇2g(rr) g( ) From the expressions of … It's completely incorrect notation and it can be confusing. In electrostatics, it is a part of LaPlace's equation and Poisson's equation for relating electric potential to charge density. In rectangular coordinates: The Laplacian finds application in the Schrodinger equation in quantum mechanics. Solution for Evaluate the Laplacian of the scalar field A = e¬" sin 0 and evaluate at P (r = 2,0 = ",¢ (The given angles are in radians) Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. is called the Laplacian.The Laplacian is a good scalar operator (i.e., it is coordinate independent) because it is formed from a combination of div (another good scalar operator) and (a good vector operator).What is the physical significance of the Laplacian? First off, the Laplacian operator is the application of the divergence operation on the gradient of a scalar quantity. $$ \Delta q = \nabla^2q = \nabla . It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of DigitalCommons@UConn. \nabla q$$ Lets assume that we apply Laplacian operator to a physical and tangible scalar quantity such as the water pressure (analogous to the electric potential). laplacian(f) computes the Laplacian of the scalar function or functional expression f with respect to a vector constructed from all symbolic variables found in f.The order of … If the function is vector valued, then its Laplacian is vector valued. I abhor the del squared notation that you've used for this reason. In one dimension, reduces to .Now, is positive if is concave (from above) and negative if it is convex.