To derive the Helmholtz theorem start first with one representation of the delta-function in 3-dimensions<\/p>\n
\\[ \\nabla^2 \\left( \\frac{1}{|\\vec r – \\vec r \\, {}’|} \\right) = -4 \\pi \\delta( \\vec r – \\vec r \\, {}’ ) \\, .\\]<\/p>\n
Start with the identity
\n\\[ \\vec F(\\vec r) = \\int_{V} d^3 r’ \\delta(\\vec r – \\vec r \\,{}’) F(\\vec r \\,{}’) \\]
\nfor an arbitrary vector field $$\\vec F(\\vec r)$$ over a given volume $$V$$. Note that time will not be involved in this derivation. Also note that there is a ongoing discussion in the literature about the correct way to extend this theorem for time varying fields. This will be a discussed in a future post.<\/p>\n
Using the explicit representation of the delta-function stated above and factoring out the derivatives with respect to the field point $$\\vec r$$ yields<\/p>\n
\\[ \\vec F(\\vec r) = \\frac{ -\\nabla^2_{\\vec r} }{4 \\pi} \\int_V d^3 r’ \\frac{\\vec F(\\vec r\\,{}’)}{|\\vec r – \\vec r\\,{}’|} \\, . \\]<\/p>\n
Now apply the vector identity $$\\nabla^2 = \\nabla( \\nabla \\cdot ) – \\nabla \\times (\\nabla \\times)$$. Doing so allows the expression for $$\\vec F(\\vec r)$$ to take the form
\n\\[ \\vec F (\\vec r) = \\frac{1}{4 \\pi} \\nabla_{\\vec r} \\times \\vec I_{vector} – \\frac{1}{4 \\pi} \\nabla_{\\vec r} I_{scalar} \\]
\nwhere the integrals
\n\\[ \\vec I_{vector} = \\nabla_{\\vec r} \\times \\int_v d^3 r ‘ \\frac{\\vec F (\\vec r\\,’)}{|\\vec r – \\vec r\\,’|} \\]
\nand
\n\\[ I_{scalar} = \\nabla_{\\vec r} \\cdot \\int_v d^3 r ‘ \\frac{\\vec F (\\vec r\\,’)}{|\\vec r – \\vec r\\,’|} \\; . \\]<\/p>\n
The strategy for handling these terms is to<\/p>\n
Application of this strategy to the vector (first) integral gives
\n\\[ \\vec I _{vector} = \\int_V d^3 r’ \\frac{ \\nabla_{\\vec r\\,’} \\times \\vec F (\\vec r \\, ‘)}{|\\vec r – \\vec r\\,’|} – \\int_{\\partial V} dS \\frac{\\hat n \\times \\vec F(\\vec r\\,’)}{|\\vec r – \\vec r\\,’|} \\; .\\]<\/p>\n
Likewise, the same strategy applied to the scalar (second) integral gives
\n\\[ I_{scalar} = \\int_V d^3 r’ \\frac{ \\nabla \\cdot \\vec F ( \\vec r \\,’)}{|\\vec r – \\vec r\\,’|} – \\int_{\\partial V} dS \\frac{\\hat n \\cdot \\vec F ( \\vec r \\,’)}{|\\vec r – \\vec r\\,’|} \\; . \\]<\/p>\n
Now the usual case of interest sets the bounding volume to be all space, which requires that the field drop off faster than $$r^{-1}$$. If this condition is met then the surface integrals zero and the original field can be written as
\n\\[ \\vec F (\\vec r) = -\\nabla U(\\vec r) + \\nabla \\times \\vec W(\\vec r) \\]
\nwhere
\n\\[ U(\\vec r) = \\frac{1}{4\\pi} \\int_V d^3 r’ \\frac{ \\nabla’ \\cdot \\vec F ( \\vec r \\,’)}{|\\vec r – \\vec r\\,’|} \\]
\nand
\n\\[ \\vec W(\\vec r) = \\frac{1}{4\\pi} \\int_V d^3 r’ \\frac{ \\nabla’ \\times \\vec F ( \\vec r \\,’)}{|\\vec r – \\vec r\\,’|} \\; .\\]<\/p>\n
At this point it is a snap to derive Coulomb’s and Biot-Savart’s laws from the Maxwell equations but that is a post for another time.<\/p>\n","protected":false},"excerpt":{"rendered":"
To derive the Helmholtz theorem start first with one representation of the delta-function in 3-dimensions \\[ \\nabla^2 \\left( \\frac{1}{|\\vec r – \\vec r \\, {}’|} \\right) = -4 \\pi \\delta(… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[4,2,3],"class_list":["post-27","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-delta-function","tag-em","tag-helmholtz"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/27","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=27"}],"version-history":[{"count":39,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/27\/revisions"}],"predecessor-version":[{"id":104,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/27\/revisions\/104"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=27"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=27"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}