{"id":5638,"date":"2014-08-04T06:05:57","date_gmt":"2014-08-03T21:05:57","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5638"},"modified":"2014-07-02T11:13:12","modified_gmt":"2014-07-02T02:13:12","slug":"tranformations-of-multiple-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5638","title":{"rendered":"TRANFORMATIONS OF MULTIPLE INTEGRALS"},"content":{"rendered":"
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In evaluating a multiple integral over a region \\cal R, it is often convenient to use coordinates other than rectangular, such as the curvilinear coordinates considered in Chapter 5. <\/p>\n

If we let (u, v) be curvilinear coordinates of points in a plane, there will be a set of transformation equations x = f(u, v),\\ y = g(u, v) mapping points (x, y) of the xy<\/em> plane into points (u, v) of the uv<\/em> plane. In such case the region \\cal R of the xy<\/em> plane is mapped into a region {\\cal R}' of the uv<\/em> plane. We then have<\/p>\n

\\displaystyle \\underset{\\cal R}{\\iint}F(x, y)dxdy = \\underset{{\\cal R}'}{\\iint} G(u, v)\\left|\\frac{\\partial (x,y)}{\\partial (u, v)}\\right| dudv \\cdots(9) <\/p>\n

where G(u, v) \\equiv F\\{f(u,v), g(u,v)\\} and <\/p>\n

\\displaystyle \\frac{\\partial (x, y)}{\\partial (u, v)} \\equiv \\left| \\begin{array}{cc} \\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} \\\\ \\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v} \\end{array} \\right| \\cdots (10)<\/p>\n

is the Jacobian<\/em> of x<\/em> and y<\/em> with respect to u<\/em> and v<\/em>. <\/p>\n

Similarly if (u, v, w) are curvilinear coordinates in three dimensions, there will be a set of transformation equations x = f(u, v, w), y = g(u, v, w), z = h(u, v, w) and we can write<\/p>\n

\\displaystyle \\underset{\\cal R}{\\iiint}F(x, y, z)dxdydz = \\underset{{\\cal R}'}{\\iiint} G(u, v, w) \\left| \\frac{\\partial (x, y, z)}{\\partial (u, v, w)} \\right|dudvdw \\cdots(11)<\/p>\n

where G(u, v, w) \\equiv F\\{f(u, v, w), g(u, v, w), h(u, v, w)\\} and<\/p>\n

\\displaystyle \\frac{\\partial (x, y, z)}{\\partial (u, v, w)} \\equiv \\left| \\begin{array}{ccc} \\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} & \\frac{\\partial x}{\\partial w} \\\\ \\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v} & \\frac{\\partial y}{\\partial w} \\\\ \\frac{\\partial z}{\\partial u} & \\frac{\\partial z}{\\partial v} & \\frac{\\partial z}{\\partial w} \\end{array} \\right|\\cdots(12)<\/p>\n

is the Jacobian<\/em> of x<\/em>, y<\/em> and z<\/em> with respect to u<\/em>, v<\/em> and w<\/em>. <\/p>\n

The results (9) and (11) correspond to change of variables for double and triple integrals. <\/p>\n

Generalizations to higher dimensions are easily made. <\/p>\n<\/blockquote>\n