{"id":22809,"date":"2023-03-01T02:16:25","date_gmt":"2023-03-01T02:16:25","guid":{"rendered":"https:\/\/www.booksofall.com\/pl\/?post_type=product&p=22809"},"modified":"2023-03-01T02:16:25","modified_gmt":"2023-03-01T02:16:25","slug":"the-zakon-series-on-mathematical-analysis-mathematical-analysis-volume-ii","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/pl\/the-zakon-series-on-mathematical-analysis-mathematical-analysis-volume-ii\/","title":{"rendered":"The Zakon Series on Mathematical Analysis – Mathematical Analysis Volume II"},"content":{"rendered":"
\n

Chapter 6 – Differentiation on \"\"\u00a0and Other Normed Linear Spaces<\/h2>\n

1. Directional and Partial Derivatives<\/strong><\/p>\n

In Chapter 5 we considered functions f : E1 \u2192 E of one real variable<\/a>.<\/p>\n

Now we take up functions f : E\u2032 \u2192 E where both E\u2032 and E are normed spaces.<\/p>\n

The scalar field of both is always assumed the same: E1 or C (the complex field<\/a>). The case E = E\u2217 is excluded here; thus all is assumed finite.<\/p>\n

We mostly use arrowed letters \"\" for vectors in the domain space E\u2032, and nonarrowed letters for those in E and for scalars.<\/p>\n

As before, we adopt the convention that f is defined on all of E\u2032, with f(~x) = 0 if not defined otherwise.<\/p>\n

Note that, if ~p \u2208<\/a> E\u2032, one can express any point \"\" as<\/p>\n

\"\"<\/p>\n

with t \u2208 E1 and ~u a unit vector<\/a>. For if \"\" set<\/p>\n

\"\"<\/p>\n

and if \"\"\u00a0set t = 0, and any ~u will do. We often use the notation<\/p>\n

\"\"<\/p>\n

First of all, we generalize Definition 1 in Chapter 5,<\/p>\n

Definition 1<\/strong><\/p>\n

Given \"\", we define the directional derivative<\/a> of f along ~u (or ~u-directed derivative of f) at ~p by<\/p>\n

\"\"<\/p>\n

if this limit exists in E (finite). We also define the ~u-directed derived function,<\/span><\/p>\n<\/div>\n

\n

\"\"<\/p>\n

as follows. For any \"\"<\/p>\n

\"\"\u00a0 if this limit exists, <\/span>otherwise.<\/p>\n

Thus \"\" is always defined, but the name derivative is used only if the limit (1) exists (finite). If it exists for each \"\" in a set B \u2286 E\u2032, we call \"\" (in classical notation \u2202f\/\u2202~u) the ~u-directed derivative of f on B.<\/p>\n

Note that, as t \u2192 0, \"\" tends to ~p over the line \"\" Thus \"\" can be treated as a relative limit over that line. Observe that it depends on both the direction and the length of\u00a0\"\". Indeed, we have the following result.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"