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The symbol $\Delta u$ represents the Laplacian of the scalar field $u$. It is a compact notation for the sum of the unmixed second partial derivatives of $u$ with respect to all coordinate directions.

Conditions: $u$ is a scalar function of multiple variables.; The coordinates are orthogonal Cartesian coordinates (implied by the simple sum of second derivatives).; The notation generalizes the 2D case to $n$ dimensions.

The symbol $\Delta u$ represents the Laplacian of the scalar field $u$. It is a compact notation for the sum of the unmixed second partial derivatives of $u$ with respect to all coordinate directions.

Conditions: $u$ is a scalar function of multiple variables.; The coordinates are orthogonal Cartesian coordinates (implied by the simple sum of second derivatives).; The notation generalizes the 2D case to $n$ dimensions.

The upside-down triangle symbol, known as nabla ($\nabla$), represents the del operator. In the context of Laplace's equation, its square ($\nabla^2$) is equivalent to the Laplacian operator ($\Delta$).

Conditions: The symbol $\nabla$ is the vector differential operator.; $\nabla^2$ denotes the composition of the divergence of the gradient, resulting in a scalar operator.; The equation is the Laplace equation.

The upside-down triangle symbol, known as nabla ($\nabla$), represents the del operator. In the context of Laplace's equation, its square ($\nabla^2$) is equivalent to the Laplacian operator ($\Delta$).

Conditions: The symbol $\nabla$ is the vector differential operator.; $\nabla^2$ denotes the composition of the divergence of the gradient, resulting in a scalar operator.; The equation is the Laplace equation.