Exam 18: Differential Forms and Exterior Calculus

Let = 2dx + 5dy, = -dx + 7dy and $\theta$ = -3dx + c dy be 1-forms on . Find the real number c such that 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 11ee7bbb_3770_208d_ae82_59bc8eff822f_TB9661_11 = 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11 $\theta$ .

Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain . (a) Is 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 closed? (b) Is 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 exact on D? If so, find a differential 1-form such that 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 = d11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 .

Find d (x), where M is the 2-manifold in given parametrically by , sin(2 ), 3 ) for 0 ≤ ≤ 1, 0 ≤ ≤ 1.

Let M be the smooth 2-manifold , x = p( $\theta$ , ) = (cos( $\theta$ )sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), sin( $\theta$ )sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), cos(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ),0 $\le$$\theta$ $\le$ 2 $\pi$ , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.

Which of the following is an antiderivative of (3x + 4xy) dx dy?

Let F(x, y) and G(x, y) be differential 0-forms on a domain D in . Prove that(dF)∧(dG) = dx∧dy.

Find the dimension of ( ).

Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R⁴ ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , .

Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to

You probably know by now that a differential k-form k $\ge$ 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in .

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

Let be a differential k-form and be a differential -form on a domain D and letd(11ee7bbd_9e7e_0bb6_ae82_bb619a089092_TB9661_11 11ee7bbd_b7c7_e467_ae82_c37bb85a20c0_TB9661_11 ) (D). Express m in terms of k and l.

Evaluate the integral of = d d + d 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d + d 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d over the upper hemispherical surface + + = 16, $\ge$ 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing.

Find the 2-volume of the 2-manifold in given parametrically by for 0 $\le$ u $\le$ 1, 0 $\le$ v $\le$ $\pi$ .

Let Φ = (2xy - ) dx + (2yz + ) dy + ( - 2zx) dz be a differential 1-form defined on a star-like domain D in . (a) Show that Φ is exact on D. (b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

The 2-manifold M in R⁴ given by the equations 0 < x₄ < 1, 0 < < 1 has normals It is oriented by the 2-form ω(x)( v₁ , v₂ ) = det( n₁ n₂ v₁ v₂ ). Let be a parametrization for M. Which of the following statements is true?

Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz.

Let ( ) be the vector space of all 3-forms on and ( ) be the vector space of all 5-forms on . If ( ) and ( ) have the same dimension, find n.

Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dx - 9z dx11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dy and D is the oriented boundary of the conical domain D = .

If is a differential k-form, where k $\ge$ 1 is an even integer, then d(11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 d11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 ) = .