{"id":147,"date":"2018-11-15T04:28:56","date_gmt":"2018-11-15T04:28:56","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=147"},"modified":"2018-11-15T08:40:10","modified_gmt":"2018-11-15T08:40:10","slug":"vector-integration","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/vector-integration\/","title":{"rendered":"Vector integration"},"content":{"raw":"
\r\n

TABLE OF CONTENTS<\/strong><\/p>\r\n

1. Ordinary integration of vectors<\/p>\r\n

2. Line integral<\/p>\r\n

2.1 Evaluation of line integral<\/p>\r\n

2.2 Conservative fields<\/p>\r\n

3. Surface integrals<\/p>\r\n

4. Volume integrals<\/p>\r\n

5. Fundamental theorems of vector calculus<\/p>\r\n

5.1 Fundamental theorem for divergence \u2013 Gauss\u2019 theorem<\/p>\r\n

5.2 Green\u2019s theorem in a plane<\/p>\r\n

5.3 Fundamental theorem for curl - Stokes\u2019 theorem<\/p>\r\n

5.4 Some other important theorems<\/p>\r\n \r\n\r\n<\/div>\r\n

LEARNING OBJECTIVES<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. In this module the topic of vector integration is taken up. First the ordinary integration of a vector is described.<\/span><\/li>\r\n \t
  2. Next the central concept of line integral is introduced. Evaluation of line integrals is described by taking up examples.<\/span><\/li>\r\n \t
  3. The special case of line integral of conservative fields is described in details with examples.<\/span><\/li>\r\n \t
  4. Next surface integral of vectors is described both for the case of open and closed surface.<\/span><\/li>\r\n \t
  5. Evaluation of volume integral is explained by an example.<\/span><\/li>\r\n \t
  6. Lastly certain fundamental theorems regarding the line, volume and surface integrals are described. The Gauss theorem, Green\u2019s theorem in a plane and the Stoke\u2019s theorem are enunciated and proved. Some other fundamental theorems are also stated without proof.<\/span><\/li>\r\n<\/ol>\r\n

    V e c t o r I n t e g r a t i o n<\/strong><\/p>\r\n\r\n

    \r\n\r\n \r\n\r\n1. Ordinary integration of vectors<\/span><\/strong>\r\n\r\n \r\n

    After having studied differentiation of vectors and vector fields, our next task is to study vector integration. The most useful concepts in this regard are the line, surface<\/em> and volume integrals<\/em> of vector fields. We first define the ordinary derivative of a vector quantity. If \u20d7 is a vector function of a single scalar variable u<\/em>, its integral over u<\/em> is defined like the integral of a function of one variable. Let<\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

    We now look on the integrals of special interest to us, viz., the line, surface<\/em> and volume integrals<\/em> of vector fields<\/em>, and also of scalar<\/em> and tensor fields<\/em>. The line, surface and volume integrals refer to integral of a field over a curve, a surface or a volume in the three-dimensional space.<\/p>\r\n \r\n\r\n2. Line integral<\/strong><\/span>\r\n\r\n<\/div>\r\n\"\"\r\n\r\n\"\"\r\n

    \r\n

    2.1 Evaluation of line integral<\/span><\/strong><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    <\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n2.2 Conservative fields<\/strong>\r\n\r\nIn analogy with the force, any vector field whose line integral is independent of the chosen path and depends only on the two end points is called a conservative field<\/em>. Both the above examples are of non-conservative fields<\/em>. In fact there is a simple and well known criterion to decide whether a given field is conservative or not. The result is given by the following theorems:\r\n\r\n \r\n\r\nTheorem-1<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\nHence\r\n\r\n\"\"without any reference to the path taken.<\/span>\r\n

    Theorem-2<\/span><\/p>\r\n\"\"Now we know from the theorem of vector differentiation that a vector field can be written as gradient of a scalar, if and only if, its curl vanishes:<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\nThus we have a simple criterion.\r\n\r\n \r\n\r\nTheorem-3<\/span>\r\n\r\nThe line integral of a vector field over a closed curve is zero, if and only if, the field is curl free.\r\n\r\n \r\n\r\nExample\r\n\r\nProve that the integral\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n3.\u00a0 Surface integrals<\/span><\/strong><\/span>\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n

    If the integration is over a closed surface, it is usually denoted by \u222f . In general we would expect the integral over a surface to depend on the boundary as well as the actual surface with that boundary. However there is a class of functions for which the integral depends only on the boundary and not the actual surface. For such functions the integral over a closed surface is zero.<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\nIf the projection of the surface S<\/em> on the x-y<\/em> plane is the region R<\/em>, then the given surface integral takes the form\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n4. Volume integrals<\/strong><\/span>\r\n

    Consider a closed surface in space enclosing a volume V<\/em>.\u00a0 Then volume integral is an expression of the form<\/p>\r\n\"\"\"\"Example<\/span><\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n

    In this case the integration can be performed in any order.\u00a0 So we write<\/p>\r\n\"\"\r\n\r\n \r\n\r\n5. Fundamental theorems of vector calculus<\/strong><\/span>\r\n

    The fundamental theorem of calculus states that<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\r\n

    The total change in function will be<\/div>\r\n
    \"\"This may be regarded as the fundamental theorem for gradients.<\/span><\/div>\r\n
    \r\n\r\n \r\n\r\n5.1 Fundamental theorem for divergence \u2013 Gauss\u2019 theorem<\/strong>\r\n\r\nThe fundamental theorem for divergence states that\r\n\r\n\"\"\r\n

    \u00a0 This theorem is most often referred to as Gauss theorem<\/em> and sometimes as Greens theorem<\/em>. The \u201cboundary\u201d of a curve is its end points, that of an open surface is its perimeter and that of volume is the enclosing surface. This theorem is also in the spirit of the fundamental theorem of calculus in that it relates the integral of the derivative of a function over a volume to the function at its boundary, that is, the bounding surface.<\/p>\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n<\/div>\r\n

    \r\n

    Let S<\/em> be a convex closed surface. Any line parallel to one of the axes will cut the surface in at most two points. In the case of line being parallel to the z<\/em>-axis, such points will divide the surface into two parts, the lower and the upper part. Let the equations of the upper and lower parts be respectively<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\nSo that\r\n\r\n\"\"\r\n\r\nSimilarly on projecting the give surface on the other two coordinate planes and adding all the contributions together, we obtain\r\n\r\n\"\"\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\nExample<\/span>\r\n\r\n\"\"\r\n\r\n5.2 Green\u2019s theorem in a plane<\/span><\/strong>\r\n\r\n\"\"\r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\nOn adding the two results we obtain the desired result:\r\n\r\n\"\"\r\n\r\n \r\n\r\n5.3 Fundamental theorem for curl - Stokes\u2019 theorem<\/strong>\r\n

    We now take up the Stokes\u2019 theorem. This theorem is also in the mould of the fundamental theorem of calculus. It relates the surface integral of the derivative of a function to the value of the function along the boundary of the surface, i.e., its periphery. The theorem states that<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\nProof of Stokes\u2019 theorem<\/span>\r\n

    \r\n

    We now come to the proof of Stokes\u2019 theorem, which as we have mentioned can be regarded as the generalization of the above proved Green\u2019s theorem to surfaces in three dimensions. Let S<\/em> be a surface whose projections in the three coordinate planes are regions bounded by simple closed curves. The equation of the surface can be written in any of the three given forms<\/p>\r\n\"\"\r\n\r\nWe have to demonstrate that\r\n\r\n\"\"\r\n\r\n<\/div>\r\n\"\"\r\n

    \r\n\r\nConsider the first term\r\n\r\n\"\"\r\n\r\n\"\"\r\n

    Hence equation (15) becomes<\/p>\r\n\"\"\r\n

    Using this expression in the surface integral, we have<\/p>\r\n\"\"\r\n

    Here R<\/em> is the projection of S<\/em> on the x-y<\/em> plane. Now we use the Green\u2019s theorem for a plane which we have just proved above and obtain<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n\r\n\"\"\r\n

    \r\n\r\nOn making similar projections on the other two planes and adding the results together we obtain the desired result\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\nExample<\/span>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n5.4 Some other important theorems<\/strong>\r\n

    There are quite a few other useful and related theorems which either follow from the above theorems or can be proved in very similar ways. We simply list these theorems without offering any proof.<\/p>\r\n\r\n<\/div>\r\n

    \"\"<\/div>\r\n\"\"\r\n\r\nSUMMARY<\/strong>\r\n\r\n \r\n
      \r\n \t
    • In this module we take up the topic of vector integration. First we describe the ordinary integration of a vector.<\/li>\r\n \t
    • Next we introduce the central concept of line integral and describe the evaluation of line integrals by examples.<\/li>\r\n \t
    • We discuss in detail the special case of line integral of conservative fields and prove theorems on the condition for a field to be conservative.<\/li>\r\n \t
    • After line integral we describe surface integrals of vectors both for the case of open and closed surfaces.<\/li>\r\n \t
    • Then we explain evaluation of volume integral by an example.<\/li>\r\n \t
    • We next state and prove the very important fundamental theorems regarding the line, volume and surface integrals; viz., the Gauss theorem, Green\u2019s theorem in a plane and the Stoke\u2019s theorem.<\/li>\r\n \t
    • Finally we state some other useful theorems without proof.<\/li>\r\n<\/ul>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n20\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\nCrystallography & crystal growth<\/strong><\/td>\r\n<\/tr>\r\n
      <\/td>\r\nMaterial science<\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\nExperimental methods for x-ray diffraction<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n ","rendered":"
      \n

      TABLE OF CONTENTS<\/strong><\/p>\n

      1. Ordinary integration of vectors<\/p>\n

      2. Line integral<\/p>\n

      2.1 Evaluation of line integral<\/p>\n

      2.2 Conservative fields<\/p>\n

      3. Surface integrals<\/p>\n

      4. Volume integrals<\/p>\n

      5. Fundamental theorems of vector calculus<\/p>\n

      5.1 Fundamental theorem for divergence \u2013 Gauss\u2019 theorem<\/p>\n

      5.2 Green\u2019s theorem in a plane<\/p>\n

      5.3 Fundamental theorem for curl – Stokes\u2019 theorem<\/p>\n

      5.4 Some other important theorems<\/p>\n

       <\/p>\n<\/div>\n

      LEARNING OBJECTIVES<\/strong><\/p>\n

        \n
      1. In this module the topic of vector integration is taken up. First the ordinary integration of a vector is described.<\/span><\/li>\n
      2. Next the central concept of line integral is introduced. Evaluation of line integrals is described by taking up examples.<\/span><\/li>\n
      3. The special case of line integral of conservative fields is described in details with examples.<\/span><\/li>\n
      4. Next surface integral of vectors is described both for the case of open and closed surface.<\/span><\/li>\n
      5. Evaluation of volume integral is explained by an example.<\/span><\/li>\n
      6. Lastly certain fundamental theorems regarding the line, volume and surface integrals are described. The Gauss theorem, Green\u2019s theorem in a plane and the Stoke\u2019s theorem are enunciated and proved. Some other fundamental theorems are also stated without proof.<\/span><\/li>\n<\/ol>\n

        V e c t o r I n t e g r a t i o n<\/strong><\/p>\n

        \n

         <\/p>\n

        1. Ordinary integration of vectors<\/span><\/strong><\/p>\n

         <\/p>\n

        After having studied differentiation of vectors and vector fields, our next task is to study vector integration. The most useful concepts in this regard are the line, surface<\/em> and volume integrals<\/em> of vector fields. We first define the ordinary derivative of a vector quantity. If \u20d7 is a vector function of a single scalar variable u<\/em>, its integral over u<\/em> is defined like the integral of a function of one variable. Let<\/p>\n<\/div>\n

        \n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        We now look on the integrals of special interest to us, viz., the line, surface<\/em> and volume integrals<\/em> of vector fields<\/em>, and also of scalar<\/em> and tensor fields<\/em>. The line, surface and volume integrals refer to integral of a field over a curve, a surface or a volume in the three-dimensional space.<\/p>\n

         <\/p>\n

        2. Line integral<\/strong><\/span><\/p>\n<\/div>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \n

        2.1 Evaluation of line integral<\/span><\/strong><\/p>\n<\/div>\n

        \n

        \"\"<\/p>\n<\/div>\n

        <\/div>\n
        \n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        2.2 Conservative fields<\/strong><\/p>\n

        In analogy with the force, any vector field whose line integral is independent of the chosen path and depends only on the two end points is called a conservative field<\/em>. Both the above examples are of non-conservative fields<\/em>. In fact there is a simple and well known criterion to decide whether a given field is conservative or not. The result is given by the following theorems:<\/p>\n

         <\/p>\n

        Theorem-1<\/span><\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        Hence<\/p>\n

        \"\"without any reference to the path taken.<\/span><\/p>\n

        Theorem-2<\/span><\/p>\n

        \"\"Now we know from the theorem of vector differentiation that a vector field can be written as gradient of a scalar, if and only if, its curl vanishes:<\/span><\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        Thus we have a simple criterion.<\/p>\n

         <\/p>\n

        Theorem-3<\/span><\/p>\n

        The line integral of a vector field over a closed curve is zero, if and only if, the field is curl free.<\/p>\n

         <\/p>\n

        Example<\/p>\n

        Prove that the integral<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

         <\/p>\n

        3.\u00a0 Surface integrals<\/span><\/strong><\/span><\/p>\n<\/div>\n

        \n

        \"\"<\/p>\n

        If the integration is over a closed surface, it is usually denoted by \u222f . In general we would expect the integral over a surface to depend on the boundary as well as the actual surface with that boundary. However there is a class of functions for which the integral depends only on the boundary and not the actual surface. For such functions the integral over a closed surface is zero.<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        \"\"<\/p>\n

        If the projection of the surface S<\/em> on the x-y<\/em> plane is the region R<\/em>, then the given surface integral takes the form<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        4. Volume integrals<\/strong><\/span><\/p>\n

        Consider a closed surface in space enclosing a volume V<\/em>.\u00a0 Then volume integral is an expression of the form<\/p>\n

        \"\"\"\"Example<\/span><\/span><\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        In this case the integration can be performed in any order.\u00a0 So we write<\/p>\n

        \"\"<\/p>\n

         <\/p>\n

        5. Fundamental theorems of vector calculus<\/strong><\/span><\/p>\n

        The fundamental theorem of calculus states that<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \"\"<\/p>\n

        The total change in function will be<\/div>\n
        \"\"This may be regarded as the fundamental theorem for gradients.<\/span><\/div>\n
        \n

         <\/p>\n

        5.1 Fundamental theorem for divergence \u2013 Gauss\u2019 theorem<\/strong><\/p>\n

        The fundamental theorem for divergence states that<\/p>\n

        \"\"<\/p>\n

        \u00a0 This theorem is most often referred to as Gauss theorem<\/em> and sometimes as Greens theorem<\/em>. The \u201cboundary\u201d of a curve is its end points, that of an open surface is its perimeter and that of volume is the enclosing surface. This theorem is also in the spirit of the fundamental theorem of calculus in that it relates the integral of the derivative of a function over a volume to the function at its boundary, that is, the bounding surface.<\/p>\n

        \"\"<\/p>\n

         <\/p>\n

        Proof<\/span><\/p>\n<\/div>\n

        \n

        Let S<\/em> be a convex closed surface. Any line parallel to one of the axes will cut the surface in at most two points. In the case of line being parallel to the z<\/em>-axis, such points will divide the surface into two parts, the lower and the upper part. Let the equations of the upper and lower parts be respectively<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        So that<\/p>\n

        \"\"<\/p>\n

        Similarly on projecting the give surface on the other two coordinate planes and adding all the contributions together, we obtain<\/p>\n

        \"\"\"\"<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        Example<\/span><\/p>\n

        \"\"<\/p>\n

        5.2 Green\u2019s theorem in a plane<\/span><\/strong><\/p>\n

        \"\"<\/p>\n

        Proof<\/span><\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        On adding the two results we obtain the desired result:<\/p>\n

        \"\"<\/p>\n

         <\/p>\n

        5.3 Fundamental theorem for curl – Stokes\u2019 theorem<\/strong><\/p>\n

        We now take up the Stokes\u2019 theorem. This theorem is also in the mould of the fundamental theorem of calculus. It relates the surface integral of the derivative of a function to the value of the function along the boundary of the surface, i.e., its periphery. The theorem states that<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n<\/div>\n

        Proof of Stokes\u2019 theorem<\/span><\/p>\n

        \n

        We now come to the proof of Stokes\u2019 theorem, which as we have mentioned can be regarded as the generalization of the above proved Green\u2019s theorem to surfaces in three dimensions. Let S<\/em> be a surface whose projections in the three coordinate planes are regions bounded by simple closed curves. The equation of the surface can be written in any of the three given forms<\/p>\n

        \"\"<\/p>\n

        We have to demonstrate that<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \"\"<\/p>\n

        \n

        Consider the first term<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        Hence equation (15) becomes<\/p>\n

        \"\"<\/p>\n

        Using this expression in the surface integral, we have<\/p>\n

        \"\"<\/p>\n

        Here R<\/em> is the projection of S<\/em> on the x-y<\/em> plane. Now we use the Green\u2019s theorem for a plane which we have just proved above and obtain<\/p>\n<\/div>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \n

        On making similar projections on the other two planes and adding the results together we obtain the desired result<\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        Example<\/span><\/p>\n

        \"\"<\/p>\n<\/div>\n

        \n

        \"\"<\/p>\n

        5.4 Some other important theorems<\/strong><\/p>\n

        There are quite a few other useful and related theorems which either follow from the above theorems or can be proved in very similar ways. We simply list these theorems without offering any proof.<\/p>\n<\/div>\n

        \"\"<\/div>\n

        \"\"<\/p>\n

        SUMMARY<\/strong><\/p>\n

         <\/p>\n

          \n
        • In this module we take up the topic of vector integration. First we describe the ordinary integration of a vector.<\/li>\n
        • Next we introduce the central concept of line integral and describe the evaluation of line integrals by examples.<\/li>\n
        • We discuss in detail the special case of line integral of conservative fields and prove theorems on the condition for a field to be conservative.<\/li>\n
        • After line integral we describe surface integrals of vectors both for the case of open and closed surfaces.<\/li>\n
        • Then we explain evaluation of volume integral by an example.<\/li>\n
        • We next state and prove the very important fundamental theorems regarding the line, volume and surface integrals; viz., the Gauss theorem, Green\u2019s theorem in a plane and the Stoke\u2019s theorem.<\/li>\n
        • Finally we state some other useful theorems without proof.<\/li>\n<\/ul>\n

           <\/p>\n

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           <\/p>\n

          20<\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n\n\n\n\n\n\n
          <\/td>\n<\/td>\n<\/td>\n<\/td>\nCrystallography & crystal growth<\/strong><\/td>\n<\/tr>\n
          <\/td>\nMaterial science<\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
          <\/td>\n<\/td>\n<\/td>\nExperimental methods for x-ray diffraction<\/td>\n<\/tr>\n
          <\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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