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The lecture generalizes the 2D Laplace equation by replacing the specific coordinates $x$ and $y$ with a sequence of generic coordinates $x_1, x_2, \ldots, x_n$. It introduces the compact operator notation $\Delta u$ to represent the sum of second partial derivatives across all $n$ dimensions, showing that the 2D case is simply the instance where $n=2$.

Conditions: The function $u$ depends on $n$ independent variables.; The coordinates are labeled $x_1$ through $x_n$.; The operation remains the sum of unmixed second partial derivatives.

The lecture generalizes the 2D Laplace equation by replacing the specific coordinates $x$ and $y$ with a sequence of generic coordinates $x_1, x_2, \ldots, x_n$. It introduces the compact operator notation $\Delta u$ to represent the sum of second partial derivatives across all $n$ dimensions, showing that the 2D case is simply the instance where $n=2$.

Conditions: The function $u$ depends on $n$ independent variables.; The coordinates are labeled $x_1$ through $x_n$.; The operation remains the sum of unmixed second partial derivatives.

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.

In multiple dimensions, the Laplacian $\Delta u$ is defined as the sum of the unmixed second partial derivatives of the scalar function $u$ with respect to each independent coordinate variable $x_1, x_2, \ldots, x_n$. Mathematically, $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$.

Conditions: $u$ is a twice-differentiable scalar function of $n$ variables.; The coordinates $x_1, \ldots, x_n$ form an orthogonal system (implied by the simple sum).; The definition applies to any dimension $n \ge 1$.

In multiple dimensions, the Laplacian $\Delta u$ is defined as the sum of the unmixed second partial derivatives of the scalar function $u$ with respect to each independent coordinate variable $x_1, x_2, \ldots, x_n$. Mathematically, $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$.

Conditions: $u$ is a twice-differentiable scalar function of $n$ variables.; The coordinates $x_1, \ldots, x_n$ form an orthogonal system (implied by the simple sum).; The definition applies to any dimension $n \ge 1$.