Volume Integral of the Pressure Gradient and Archimedes' Principle

TL;DR

This paper explores how vector calculus theorems, like the divergence theorem, can be applied in mechanics to derive Archimedes' principle, aiming to enhance students' understanding of these tools beyond electromagnetism.

Contribution

It presents a mechanics-based approach using vector calculus to derive Archimedes' principle, serving as a pedagogical bridge for students to connect mathematical tools across physics disciplines.

Findings

Derivation of Archimedes' principle from hydrostatic equilibrium.

Application of vector calculus in a mechanics context.

Educational benefit for physics students.

Abstract

The theorems of vector analysis (divergence theorem, etc.) are typically first applied in the undergraduate physics curriculum in the context of the electromagnetic field and the differential forms of Maxwell's equations. However, these tools are analyzed in depth several courses later in the junior-senior level. I discuss here a "bridge" problem, using the language of vector calculus in a mechanics setting to understand Archimedes' principle as a consequence of hydrostatic equilibrium and the superposition of the external forces. It is my hope that this treatment will help students better integrate and understand understand these and similar vector analysis results in contexts beyond electromagnetism.