{"id":5008,"date":"2014-03-21T06:05:42","date_gmt":"2014-03-20T21:05:42","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5008"},"modified":"2014-08-01T18:45:03","modified_gmt":"2014-08-01T09:45:03","slug":"limits-continuity-and-derivatives-of-vector-functions","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5008","title":{"rendered":"Limits, continuity and derivatives of vector functions"},"content":{"rendered":"
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Limits, continuity and derivatives of vector functions follow rules similar to those for scalar functions already considered. The following statements show the analogy which exists. <\/p>\n

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  1. The vector function \\bold{A}(u) is said to be continuous<\/em> at u_0 if given any positive number \\varepsilon, we can find some positive number \\delta such that \\left|\\bold{A}(u) - \\bold{A}(u_0)\\right| < \\varepsilon[\/latex] whenever [latex]\\left|u - u_0\\right| < \\delta[\/latex]. This is equivalent to the statement [latex]\\lim\\limits_{u \\rightarrow u_0}\\bold{A}(u) = \\bold{A}(u_0)[\/latex]. <\/li> <li>The derivative of <img src='https:\/\/s0.wp.com\/latex.php?latex&bg=T&fg=000000&s=0' alt='' title='' class='latex' \/>\\bold{A}(u) is defined as
    \n\\displaystyle \\frac{d\\bold{A}}{du} = \\lim\\limits_{\\Delta{u} \\rightarrow 0}\\frac{\\bold{A}(u + \\Delta {u}) - \\bold{A}(u)}{\\Delta{u}}\\cdots (7)
    \nprovided this limit exists. In case \\bold{A}(u) = A_1(u)\\bold{i} + A_2(u)\\bold{j} + A_3(u)\\bold{k}; then
    \n\\displaystyle \\frac{d\\bold{A}}{du} = \\frac{dA_1}{du}\\bold{i} + \\frac{dA_2}{du}\\bold{j} + \\frac{dA_3}{du}\\bold{k}
    \nHigher derivatives such as d^2\\bold{A}\/du^2, etc., can be similarly defined. <\/li>\n
  2. If \\bold{A}(x, y, z) = A_1(x, y, z)\\bold{i} + A_2(x, y, z)\\bold{j} + A_3(x, y, z)\\bold{k}, then
    \n\\displaystyle d\\bold{A} = \\frac{\\partial\\bold{A}}{\\partial x}dx + \\frac{\\partial\\bold{A}}{\\partial y}dy + \\frac{\\partial\\bold{A}}{\\partial z}dz\\cdots(8)
    \nis the differential<\/em> of \\bold{A}. <\/li>\n
  3. Derivatives of products obey rules similar to those for scalar functions. However, when cross products are involved the order may be important. Some examples are:
    \n\\displaystyle (a)\\ \\frac{d}{du}(\\phi\\bold{A}) = \\phi\\frac{d\\bold{A}}{du} + \\frac{d\\phi}{du}\\bold{A}
    \n\\displaystyle (b)\\ \\frac{\\partial}{\\partial y}(\\bold{A} \\cdot \\bold{B}) = \\bold{A} \\cdot \\frac{\\partial \\bold{B}}{\\partial y} + \\frac{\\partial\\bold{A}}{\\partial y} \\cdot \\bold{B}
    \n\\displaystyle (c)\\ \\frac{\\partial}{\\partial z}(\\bold{A} \\times \\bold{B}) = \\bold{A} \\times \\frac{\\partial\\bold{B}}{\\partial z} + \\frac{\\partial\\bold{A}}{\\partial z} \\times \\bold{B}<\/li>\n<\/ol>\n<\/blockquote>\n