I can't see why. I am only struggling with the last term on the right side, which is a vector Laplacian: $$\nabla^2 \mathbf{u}$$ (1) In tensor notation, with the aid of the covariant derivative, this can be written as:

of Kansas Dept. Hot Network Questions Is there a way to make a five sided bolt out of a 32 sided disk opening? The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. 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. Technically the unit "vectors" referred to in this tutorial are actually vector fields, since the unit vectors of a coordinate system are defined at all points in space (other than zero, at least). 1. 1. of EECS The scalar Laplacian can likewise be expressed in cylindrical and spherical coordinates; results given on page 53 of your book. I calculated the Vector Laplacian of a uniform vector field in Cartesian and in Cylindrical coordinates.
The Vector Laplacian The vector Laplacian, denoted as ∇2A(r), both operates on a vector field and results in a vector field, and is defined as: Divergence from covariant derivative?

But, for deriving Divergence in Cylindrical and Spherical, I am going to explain with another approach discussed below. We know that the divergence of the vector field is given as

Figure 2: Vector and integral identities. So we're interested now in the divergences these fields in order to complete the previous equation.

Derivation of Vector Laplacian in Cylindrical Coordinates through Tensor Analysis.

The x, y and z components of the vector are equivalently written in terms of ρ, φ and z components. Laplacian As of Version 9.0, vector analysis functionality is built into the Wolfram Language gives the Laplacian, ∇ 2 f , of the scalar function or vector field f in the default coordinate system.
9/16/2005 The Laplacian.doc 2/2 Jim Stiles The Univ. Here is a scalar function and A;a;b;c are vector elds.

Here is a scalar function and A is a vector eld. I found different results.

In Cartesian coordinates the vector field is: (vx,vy,vz)=(1,0,0). Its Laplacian is: (0,0,0) . That's the result I expected.

I'm currently trying to derive the Navier-Stokes equations in cylindrical coordinates through tensor analysis. A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. Cylindrical coordinates in PDE. Approach 2 for deriving the Divergence in Cylindrical . Figure 1: Grad, Div, Curl, Laplacian in cartesian, cylindrical, and spherical coordinates.