How is the Laplacian defined in multiple dimensions?

Q: How is the Laplacian defined in multiple dimensions?

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, $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$.

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 $n \ge 1$ .

Reasoning, step by step

Identify the number of dimensions $n$ .
List the independent variables $x_1, x_2, \ldots, x_n$ .
For each variable $x_i$ , compute the second partial derivative $\frac{\partial^2 u}{\partial x_i^2}$ .
Sum all these second partial derivatives.
Denote the result as $\Delta u$ or $\nabla^2 u$ .
Set $\Delta u = 0$ for the Laplace equation in $n$ dimensions.

Example

The lecturer writes the general formula $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$ , explaining that one takes the double derivative on each dimension and sums them up. This generalizes the 2D case where $n=2$ .

Common misconceptions

Including mixed partial derivatives (e.g., $\frac{\partial^2 u}{\partial x_1 \partial x_2}$ ) in the sum; the Laplacian only includes pure second derivatives.
Assuming the definition changes for non-Cartesian coordinates without adding metric scale factors; the simple sum applies to Cartesian coordinates.
Thinking the Laplacian is a vector; it is a scalar quantity.
Believing that $n$ must be 3; the definition holds for any $n$ .

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How is the Laplacian defined in multiple dimensions?

Conditions

Reasoning, step by step

Example

Common misconceptions

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How is the Laplacian defined in multiple dimensions?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

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