Question 1

Let  ,  ,    be 1-forms and   be a 2-form on   . Which of the following is a 3-form on   ?
A)   +   + 
B)  + 
C) ( + )   
D)     
E) All of the above

Accepted Answer

D

Question 2

Let ,  be differential 1-forms and let  be a differential 0-form on  a domain D in Which of the following is a differential 2-form on D?
A) + 
B)  
C)  - 
D)  + 
E) None of the above

Accepted Answer

B

Question 3

Find the dimension of  (  ).
A) 7
B) 840
C) 4
D) 3
E) 35

Accepted Answer

E

Question 4

Let k and n be integers such that 1  k  n. Find the dimension of the vector space of all k-forms on  .
A) 
B) n
C) k
D) n - k
E) n(n - 1)(n - 2).....(n - k +1)

Accepted Answer

The answer of Let k and n be integers such...

Question 5

Let  =  . List all basis vectors of  (  ).
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Let  =  . List all...

Question 6

If  is a k-form and  is an l-form on  , then  =  .

Accepted Answer

The answer of If  is a k-form and ...

Question 7

Let φ and ψ be two 1-forms on  say φ=  d  +  d  , ψ=  d  +  d  . Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.

Accepted Answer

The answer of Let φ and ψ be two 1-forms...

Question 8

A) u . v
B) u × v
C) u + v
D)  + 
E)

Accepted Answer

The answer of 
A) u . v
B) u × v
C)...

Question 9

If  is a k -form,   are l-forms on  , then  =  implies  =.

Accepted Answer

The answer of If  is a k -form, ...

Question 10

A) u + v
B) u - v
C) u × v
D) v × u
E) v - u

Accepted Answer

The answer of 
A) u + v
B) u - v
C)...

Question 11

L = 9dx - 2dy and  = -dx+ 3dy be 1 -forms on  . Find all 1-forms  on  such that  .
A)  = t dx +  (25 - 2t) dy, t  R
B)  = t dx +  (29 - 2t) dy, - < t < 
C)  = t dx +  (29 + 2t) dy, - < t < 
D)  =  (25 + 9t) dx + t dy, t  R
E)  =  (29 - 9t) dx + t dy, - < t <

Accepted Answer

The answer of L = 9dx - 2dy and ...

Question 12

= 0 for every k-form  on  .

Accepted Answer

The answer of  = 0 for every k-form ...

Question 13

Simplify a  dy  dz + b dy + c dx + (a + b +c) dy.
A) (a - b + c) dz
B) (a +b + c) dz
C) (a - b - c) dz
D) (a + b -c) dz
E) (a + b +c) dy

Accepted Answer

The answer of Simplify a  dy  dz +...

Question 14

Let  = 2dx + 5dy, = -dx + 7dy and $\theta$ = -3dx + c dy be 1-forms on  . Find the real number c such that     =  $\theta$.
A) 12
B) 2
C) 17
D) -3
E) 0

Accepted Answer

The answer of Let  = 2dx + 5dy, =...

Question 15

Let  ,  ,   be m-forms,  be a k-form, and   be an l-form on  . Which of the following properties of the wedge product is not always true?
A) (  +  )  =     +   
B) (   )   =  (  )
C)     = -   
D) (a  )    = a (  ) for a   R
E) for every k $\ge$  1, there exists a zero k-form such that  + 0 = , and 0       = 0.

Accepted Answer

The answer of Let  ,  ,  ...

Question 16

Let eS1U1B1iS1U1B0 , i = 1, 2, 3, 4  be the standard basis vectors in RS1U1P14S1S1P0  and let     
A) 32
B) -56
C) -40
D) -14
E) 40

Accepted Answer

The answer of Let eS1U1B1iS1U1B0 , i = 1, 2,...

Question 17

Let ψ = 12dy∧dz -10 dx∧dz + 8 dx∧dy be a 2-form on  . Express ψ as a product of two 1-forms.Hint: Add a suitable multiple of dz∧dz = 0 to ψ.

Accepted Answer

The answer of Let ψ = 12dy∧dz -10 dx∧dz +...

Question 18

Let σ = d  ∧d  + d  ∧d  be a 2-form on  . Express σ as a product of two 1-forms.

Accepted Answer

The answer of Let σ = d  ∧d ...

Question 19

Show that if φ is a k-form on  , then φ∧φ = 0 if k is odd.

Accepted Answer

The answer of Show that if φ is a k-form...

Question 20

Let   be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn( ) = -1.

Accepted Answer

The answer of Let   be the permutation that...

Quiz 18: Differential Forms and Exterior Calculus

Let , , be 1-forms and be a 2-form on . Which of the following is a 3-form on ?

Let , be differential 1-forms and let be a differential 0-form on a domain D in Which of the following is a differential 2-form on D?

Find the dimension of ( ) .

Let k and n be integers such that 1 k 11ee7973_99f4_ef7a_88d3_478f26d4adc3_TB9661_11 n. Find the dimension of the vector space of all k-forms on 11ee77e1_778d_c258_a0f8_ebae20fedc9b_TB9661_11 .

Let = . List all basis vectors of ( ) .

If is a k-form and is an l-form on , then = .

Let φ and ψ be two 1-forms on say φ= d + d , ψ= d + d . Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.

If is a k -form, and are l-forms on , then = implies =.

Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that .

= 0 for every k-form on .

Let e_i , i = 1, 2, 3, 4 be the standard basis vectors in R⁴ and let

Let ψ = 12dy∧dz -10 dx∧dz + 8 dx∧dy be a 2-form on . Express ψ as a product of two 1-forms.Hint: Add a suitable multiple of dz∧dz = 0 to ψ.

Let σ = d ∧d + d ∧d be a 2-form on . Express σ as a product of two 1-forms.

Show that if φ is a k-form on , then φ∧φ = 0 if k is odd.

Let be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn(11ee7bbc_f5df_3c93_ae82_ade676104027_TB9661_11 ) = -1.