Equations Involving the Laplacian Operator, Second Order Partial Derivatives 📂Mathematical Physics

Equations Involving the Laplacian Operator, Second Order Partial Derivatives

Explanation

Let’s call $T$ a scalar function and $\mathbf{A}$ a vector function.

Divergence of the Gradient: $\nabla \cdot (\nabla T) = \dfrac{\partial^{2} T}{\partial x^{2}} + \dfrac{\partial ^{2} T} {\partial y^{2}} + \dfrac{\partial ^{2} T}{\partial z^{2}}$
Curl of the Gradient: $\nabla \times (\nabla T)= \mathbf{0}$
Gradient of the Divergence: $\nabla (\nabla \cdot \mathbf{A} )$
Divergence of the Curl: $\nabla \cdot (\nabla \times \mathbf{A})=0$
Curl of the Curl: $\nabla \times (\nabla \times \mathbf{A})=\nabla ( \nabla \cdot \mathbf{A}) - \nabla ^{2} \mathbf{A}$

The results of Gradient and Curl are vectors, and the result of Divergence is a scalar, so there are five types of second-order derivatives in total.

Divergence of the Gradient

The divergence of the gradient is specifically named as Laplacian and is simply denoted as $\nabla^{2}$.

$$ \begin{align*} \nabla \cdot (\nabla T) &= \left( \dfrac{\partial }{\partial x} \hat{\mathbf{x}} + \dfrac{\partial}{\partial y} \hat{\mathbf{y}} + \dfrac{ \partial }{\partial z} \hat {\mathbf{z}} \right) \cdot \left( \dfrac{\partial T}{\partial x} \hat{\mathbf{x}} + \dfrac{ \partial T}{\partial y} \hat{\mathbf{y}} + \dfrac{\partial T}{\partial z} \hat{\mathbf{z}} \right) \\ &= \dfrac{\partial^{2} T}{\partial x^{2}} + \dfrac{\partial ^{2} T} {\partial y^{2}} + \dfrac{\partial ^{2} T}{\partial z^{2}} \\ &= \nabla^{2} T \end{align*} $$

The Laplacian is fundamentally an operator applied to a scalar function, but if used for a vector, it implies taking the Laplacian of each component (scalar) of the vector. Thus, the notation is used redundantly for these two types of operators. The reason for this redundancy is that there is no confusion about what $\nabla^{2}$ indicates, depending on whether the function is a scalar or a vector function. This is also referred to as a Vector Laplacian.

$$ \nabla^{2} \mathbf{A} \equiv (\nabla^{2} A_{x} ) \hat{\mathbf{x}} + (\nabla^{2} A_{y}) \hat{\mathbf{y}} + (\nabla^{2} A_{z} ) \hat{\mathbf{z}} $$

Curl of the Gradient

The curl of the gradient is always $\mathbf{0}$.

$$ \nabla \times (\nabla T) = \mathbf{0} $$

Gradient of the Divergence

The gradient of the divergence has no specific name and does not possess any particular properties. It is rarely encountered in physics and is not of great importance. Care should be taken not to confuse it with the Laplace operation.¹

Divergence of the Curl

The divergence of the curl is always $0$.

$$ \nabla \cdot ( \nabla \times \mathbf{A})=0 $$

Curl of the Curl

The curl of the curl can be represented as the sum of ‘Gradient of the Divergence’ and ‘Vector Laplacian’.

$$ \nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A}) - \nabla^{2} \mathbf{A} $$

Note that $\nabla ^{2} \mathbf{A}$ indicates the Vector Laplacian.