{"id":306,"date":"2011-01-12T00:00:10","date_gmt":"2011-01-12T05:00:10","guid":{"rendered":"http:\/\/www.fruechtetheory.com\/blog\/?p=306"},"modified":"2011-01-15T17:57:40","modified_gmt":"2011-01-15T22:57:40","slug":"gradient-of-a-scalar-potential","status":"publish","type":"post","link":"https:\/\/www.fruechtetheory.com\/blog\/2011\/01\/12\/gradient-of-a-scalar-potential\/","title":{"rendered":"Gradient of a Scalar Potential"},"content":{"rendered":"<p>In Thomas\u2019 Calculus we find the definition: \u201cIf <strong>F<\/strong> is a vector field defined on <em>D<\/em> and <strong>F<\/strong> = del <em>f<\/em> for some scalar function <em>f<\/em> on <em>D<\/em>, then <em>f<\/em> is called a potential function for <strong>F<\/strong>.\u201d ([1], pg 921)<\/p>\n<p>Kaplan states that the \u201cgravitational field is the gradient of the scalar <em>f<\/em> = kMm\/r\u201d ([2], Prob. 4, pg 180), which for General Relativity is indeed the case.\u00a0 With gamma ray energy exchange however, the scalar is <em>f<\/em> = k<sub>1<\/sub>M\/r, in form much like the scalar potential k<sub>2<\/sub>q\/r with electrodynamics.\u00a0 This tells us that both gravitational and Coulomb forces are of independent action, and the corresponding fields are gradients of scalar potentials, which can be considered as more evidence for unification.<\/p>\n<p>The vector fields we are looking at have units of N\/kg for gravity, and of course <strong>E<\/strong> = N\/C for an electric field.\u00a0 A gravitational field calculation is most applicable when the object in the field is small compared to, or far away from, the source of the field, such as Jupiter compared to the sun.<\/p>\n<p>Aside from the King James Version of the Bible, Wilfred Kaplan\u2019s <span style=\"text-decoration: underline;\">Advanced Calculus<\/span> is still my favorite book.\u00a0 By the way, we used Thomas\u2019 Calculus book at the University of Wisconsin in 1976 and 1977.<\/p>\n<p>\u00a0<\/p>\n<p>[1] Thomas, George B., as revised by Weir, Maurice D. and Hass, Joel, <span style=\"text-decoration: underline;\">Thomas\u2019 Calculus, Twelfth Edition<\/span>, Addison-Wesley of Pearson Education, Inc., 2010, 2005, 2001<\/p>\n<p>[2] Kaplan, Wilfred, <span style=\"text-decoration: underline;\">Advanced Calculus, Fifth Edition<\/span>, Addison-Wesley of Pearson Education, Inc., 2003<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Thomas\u2019 Calculus we find the definition: \u201cIf F is a vector field defined on D and F = del f for some scalar function f on D, then f is called a potential function for F.\u201d ([1], pg 921) Kaplan states that the \u201cgravitational field is the gradient of the scalar f = kMm\/r\u201d [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,7,13,11],"tags":[],"class_list":["post-306","post","type-post","status-publish","format-standard","hentry","category-astrophysics","category-classical-electrodynamics","category-general-relativity","category-newtonian-mechanics"],"_links":{"self":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/306","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/comments?post=306"}],"version-history":[{"count":4,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/306\/revisions"}],"predecessor-version":[{"id":314,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/306\/revisions\/314"}],"wp:attachment":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/media?parent=306"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/categories?post=306"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/tags?post=306"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}