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C C C C C n r n x y x x r n y s r nds ds r n ds

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※將粗框(灰底)的表格畫在答案紙上並填入對應的答案,每個答案 1 分。

6. 高斯定理 2D (6%) (a)C C C1 2 3r n ds 

1

1

3 0

: 0, 1

( , ) 0, 1 0

0

0 0

C

C n

r n x y y x s r n

y r nds ds

 

 

           

 

   

 

 

 

2

2

5 0

: 4,3

3 3 5 12 ( , ) 4,3 4 3

4 5

5 12 12

5

C

C n

x s

r n x y x y r n

y s

r nds ds

  

         

 

 

   

 

3

1 2 3

3

4 0

: 1,0

( , ) 1,0 0 0

4

0 0

0#

C

C C C

C n

r n x y x x r n

y s

r nds ds r n ds

 

            

   

 

 

(b) r dA

4 3 3 4

0 0 2

0 12 0 12#

r dA y dxdy

  

   

  

(c)C2r n ds 

 

 

2

2

5 0

: 4,3

3 3 5 12 ( , ) 4,3 4 3

4 5

5 12 12

5 #

C

C n

x s

r n x y x y r n

y s

r nds ds

  

         

 

 

   

 

(2)

(d)(0,0)

到L 線的高為多少? H 點的座標?

12 48 36 , , 5 # 25 25#

 

 

(e)說明為何國小老師告訴我們三角形面積 1

2底底 而不是1 3底底

# 2

  r

高斯定理3D (6%)

(a)S S1  2 S S3 4r n dS 

1 1

2 2

3 3

1

4 3 3 4 0 0

2

5 4 4 5 0 0

3

5 3 3 5 0 0

4

: dA

0 0 =0

: dA

0 0 =0

: dA

0 0 =0

20 15 12

: , ,

769 769 769

y

S S

z

S S

z

S S

S n kdxdy

r n dS dxdy dxdy

S n idydz

r n dS dydz dydz

S n jdxdz

r n dS dxdz dxdz

S n

 

   

 

   

 

   

 

  

   

   

   

 

 

 

 

 

 

4 4

4

1 2 3 4

S S S S

20 15 12

( , , ) , ,

769 769 769

60 60 769

2 30

769 769

30#

S S

S

r n dS x y z dS

dS r n dS

  

 

    

  

 

 





 

 

(3)

(b) r dV

 

3 5 5

4 - 3 5

4 4 3

0 0 0

3

3 30#

y y x

r dV xi yj zk dV dV

dzdxdy

       

 

  

 

  

(c)S4r n dS 

4 4

4

4

20 15 12

: , ,

769 769 769

20 15 12

( , , ) , ,

769 769 769

60 60 769 #

2 30

769 769

S S

S

S n

r n dS x y z dS

dS

 



 



 

(d)(0,0, 0) S對的高或是到點的距離4 H

?

1 1 1

1 1 1

2 2 2

20 15 12

( , , ), ( , , )

769 769 769

20 15 12

769 769 769 60

20 15 12 769

769

#

769 769

P x y z n

x y z

OH PO n n

 

底 底 底底底 底 底

(e) H 點的座標?

20 15 12

, ,

60 769 769 769 1200 900 720

(0,0,0) , ,

1 769 769 769

76 #

H 9

 

 

(f)說明為何四面體體積 1

3底面積 高 而不是 1 4底高   r 3#

參考文獻

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