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Evaluate ∫∫s(axi + byj + czk).n d S where S is the surface of the sphere x2 + y2 + z2 = 1. |
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Answer» \(\int\int_S\) (ax\(\hat i\) + by \(\hat j\) + c z\(\hat k\)) . \(\hat n\) ds = \(\int\int\int_V\) (\(\vec\bigtriangledown.\vec F\)) dV, where \(\vec F\) = ax\(\hat i\) + by \(\hat j\) + c z\(\hat k\) = \(\int\int\int_V\) (a + b + c) dx dy dz (\(\because\) dV = dx dv dz and \(\vec\bigtriangledown\) = \(\hat i\) \(\cfrac{\partial}{\partial x}\) + \(\hat j\) \(\cfrac{\partial}{\partial y}\) + \(\hat k\) \(\cfrac{\partial}{\partial z}\)) = (a + b + c) \(\int\int\int_V\) dx dy dz ( V is volume enclosed by sphere x2 + y2 + z2 = 1) = (a + b + c) \(\cfrac43\) \(\pi\) x 13 ( \(\because\) Radius of sphere x2 + y2 + z2 = 1 is r = 1) (\(\therefore\) volume = \(\cfrac43\) \(\pi\) . 13) = \(\cfrac43\) (a + b + c) \(\pi\) |
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