# Second Derivatives and Laplace Operator

JoVE Core
Physik
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JoVE Core Physik
Second Derivatives and Laplace Operator

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The first-order operators using the del operator includes the gradient, divergence, and curl.

Certain combinations of first-order operators on a scalar or a vector function yields second-order expressions.

The second order derivatives include: the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.

The curl of a gradient function and the divergence of a curl function are always zero.

The divergence of the gradient of a scalar function gives the scalar Laplace operator, or Laplacian. A Laplacian is analogous to the second-order derivative of the scalar quantities.

When the gradient of a scalar function is expressed in cylindrical and spherical coordinates, its Laplacian in cylindrical and spherical coordinates is obtained.

The gradient of a divergence function and the curl of a curl function are mathematical constructs. Lagrange's vector cross-product identity formula relates both to a vector Laplacian.

The vector Laplacian is obtained by directly applying the scalar Laplacian to each of the scalar components of a vector.

## Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.

Consider a scalar function. The curl of its gradient can be written as follows:

For a vector function, the divergence of a curl can be expressed as follows:

The curl of a gradient function and the divergence of a curl function are always zero.

The divergence of the gradient of a scalar function can be expressed as follows:

The Laplacian is analogous to the second-order differentiation of the scalar quantities. It describes physical phenomena like electric potentials and diffusion equations for heat flow.

The divergence of gradient and the curl of a curl are mathematical constructs. Lagrange's vector cross-product identity formula relates both to a vector Laplacian.

The vector Laplacian is obtained by directly applying the scalar Laplacian to each of the scalar components of a vector.