# divergence = flux / volume is independant of the limiting volume

Printable View

• April 26th 2010, 07:31 PM
maddas
divergence = flux / volume is independant of the limiting volume
In physics books, one often finds the coordinate-free definiton $\mathrm{div}\; \mathbf{f} = \lim \frac1{\mathrm{d}V} \oint \mathbf{f} \cdot \mathrm{d}\mathbf{S}$, where the surface integral is taken over the surface of a small volume element with volume $\mathrm{d}V$ as this volume goes to zero, together with the claim that the limit is "independant of the shape of the volume element" (which my book dismisses with "this claim is somewhat troublesome to prove"). What is the precise statement for the limit being independant of the "shape of the volume element"?
All times are GMT -8. The time now is 02:59 PM.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8805397152900696, "perplexity_flag": "middle"}