# 3.7 Laplacian Operator

In later chapters, we sometimes deal with problems involving multiple combinations of operations on scalars and vectors. A frequently encountered combination is the divergence of the gradient of a scalar. For a scalar function V defined in Cartesian coordinates, its gradient is

∇V=x ˆ∂V∂x+y ˆ∂V∂y+z ˆ∂V∂z=x ˆ A x+y ˆ A y+z ˆ A z=A, (3.108)

where we defined a vector A with components A x=∂V/∂x, A y=∂V/∂y, and A z=∂V/∂z. The divergence of ∇V is

∇⋅(∇V)=∇⋅A=∂A x∂x+∂A y∂y+∂A z∂z=∂2 V∂x 2+∂2 V∂y 2+∂2 V∂z 2. (3.109)

For convenience, ∇⋅(∇V) is called the Laplacian of V and is denoted by ∇2 V (the symbol ∇2 is pronounced “del square”). That is,

As we can see from Eq. (3.110), the Laplacian of a scalar function is a scalar. Expressions for ∇2 V in cylindrical and spherical coordinates are given in Appendix C.

The Laplacian of a scalar can be used to define the Laplacian of a vector. For a vector E specified in Cartesian coordinates as

E=x ˆ E x+y ˆ E y+z ˆ E z, (3.111)

the Laplacian of E is

∇2 E=∂2∂x 2+∂2∂y 2+∂2∂z 2 E=x ˆ∇2 E x+y ˆ∇2 E y+z ˆ∇2 E z. (3.112)

Thus, in Cartesian coordinates the Laplacian of a vector is a vector whose components are equal to the Laplacians of the vector components. Through direct substitution, it can be shown that

Concept Question 3-11: What do the magnitude and direction of the gradient of a scalar quantity represent?

Concept Question 3-12: Prove the validity of Eq. (3.84c) in Cartesian coordinates.

Concept Question 3-13: What is the physical meaning of the divergence of a vector field?

Concept Question 3-14: If a vector field is solenoidal at a given point in space, does it necessarily follow that the vector field is zero at that point? Explain.

Concept Question 3-15: What is the meaning of the transformation provided by the divergence theorem?

Concept Question 3-16: How is the curl of a vector field at a point related to the circulation of the vector field?

Concept Question 3-17: What is the meaning of the transformation provided by Stokes’s theorem?

Concept Question 3-18: When is a vector field “conservative”?