Evaluating on Cube S

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WMDhamnekar
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Without using the Divergence Theorem Evaluate the surface integral of boundary of the solid cube S=

My attempt:
Here we have to use the following definition of surface integral.
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Note that there will be a different outward unit normal vector to each of the six faces of the cube.
 
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Hi Dhamnekar Winod,

The formula you posted is for a real-valued function, but the function you provided is a vector field. You will need to use the formula for a surface integral of a vector field: See the Wikipedia - Surface Integrals of Vector Fields.

After that, think about how to parameterize each of the six faces of . For example, the "bottom" face can be parameterized as , , and , where ; i.e., for . Using this parameterization, calculate the integral in . Note: You will need to use the negative of the cross product to ensure you get an outward pointing normal vector to the "bottom" of . Then repeat the process for each of the other 5 faces of .
 
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  • #3
This integral will be over the 6 faces of the cube. Do each separately. One face is z= 0 for x and y each from 0 to 1. On z= 0, the vector field zk= 0k so the integral is 0. That is also true on the faces x= 0 and y= 0. On z= 1 we integrate zk= 1k so we are integrating 1 for both x and y from 0 to 1. That is 1 times the area of the square, which is also 1, so the integral is 1. This is also true on the faces x= 1 and y= 1 so the entire surface integral is 3.
 
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  • #4
HallsofIvy said:
This integral will be over the 6 faces of the cube. Do each separately. One face is z= 0 for x and y each from 0 to 1. On z= 0, the vector field zk= 0k so the integral is 0. That is also true on the faces x= 0 and y= 0. On z= 1 we integrate zk= 1k so we are integrating 1 for both x and y from 0 to 1. That is 1 times the area of the square, which is also 1, so the integral is 1. This is also true on the faces x= 1 and y= 1 so the entire surface integral is 3.
f(x,y,z) = xi + yj + zk , So,

This surface integral is for one face of the cube. So entire surface integral will be
 
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