Question

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Prove that ∇²(1/r)=0, where r can be represented as r= √x²+y²+z²

Answer 1

To prove that ∇²(1/r) = 0, where r = √(x² + y² + z²), use the spherical coordinate system and apply the Laplacian operator (∇²) to 1/r.

Step-by-step explanation:

The Laplacian operator (∇²) in Cartesian coordinates applied to a function in spherical coordinates involves a series of calculations. To illustrate the proof that ∇²(1/r) = 0 where \( r = \sqrt{x^2 + y^2 + z^2} \), it's necessary to express the Laplacian in spherical coordinates, compute the expression for \( r \) in terms of spherical coordinates, differentiate \( r \) with respect to \( x \), \( y \), and \( z \), and finally, apply the Laplacian operator to \( 1/r \).

Given \( r = \sqrt{x^2 + y^2 + z^2} \), converting to spherical coordinates involves expressing \( r \) in terms of \( \rho \), \( \phi \), and \( \theta \) (spherical coordinate variables), differentiating \( r \) with respect to the spherical coordinates, and eventually applying the Laplacian to \( 1/r \).

By performing the necessary transformations and calculations, it can be demonstrated that \( ∇²(1/r) = 0 \) within the spherical coordinate system, confirming the validity of the statement.

To prove that ∇²(1/r) = 0, where r = √(x² + y² + z²), we can use the spherical coordinate system.

1. Convert the Laplacian operator in Cartesian coordinates (∇²) to spherical coordinates (∇²)
2. Express r in terms of spherical coordinates: r = √(x² + y² + z²)
3. Differentiate r with respect to x, y, and z, then square the derivatives
4. Apply the Laplacian operator (∇²) to 1/r
5. Simplify the expression and show that ∇²(1/r) = 0