Exam 18: Differential Forms and Exterior Calculus

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.Let = F dx + G dy + H dz be a differential 1-form on a domain D and let be the vector field corresponding to 11ee7bc6_e1d1_1134_ae82_9ddb9868f737_TB9661_11 . Using this set up, find the vector differential identity corresponding to the fact 11ee7bc6_e1d1_1134_ae82_9ddb9868f737_TB9661_11 = .

Let S be a piece with boundary of a smooth 3-manifold in R⁴ (hypersurface) given by the equation = g( , , ) and let= d d 11ee7bcb_c6fc_6099_ae82_719973faadb8_TB9661_11 d . Apart from sign due to orientation of S, is equal to

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.

Calculate dx dz, where M is the surface given by z = , 0 $\le$ z $\le$ 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , )

Let = xdx + vdv, = dy dw, = dz11ee7bbf_c537_273d_ae82_09ae839f29a0_TB9661_11 dv11ee7bbf_c537_273d_ae82_09ae839f29a0_TB9661_11 du be differential forms in a domain D .Find d(11ee7bbf_ebe2_23df_ae82_0565ecd758af_TB9661_11 11ee7bc0_041d_24a0_ae82_87f021afe142_TB9661_11 11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 11ee7bc0_3408_e781_ae82_7de18ff0ad45_TB9661_11 ).

Let the differential 2-form = xdy dz + ydz11ee7bc9_e7ae_ceff_ae82_cb591a35a817_TB9661_11 dx + (1 - 2z)dx11ee7bc9_e7ae_ceff_ae82_cb591a35a817_TB9661_11 dy be defined in a star-like domain . (a) Is 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 closed? (b) Is 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 exact on D? If so, find a differential 1-form such that 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 = d11ee7bca_376e_ea71_ae82_a7b6e43e019b_TB9661_11.

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?

State the Divergence Theorem and Stokes's Theorem in 3-space, and Green's Theorem in 2-space in terms of differential forms.

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

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

Find , where M is the 2-manifold in given parametrically by for 0 < < 1, 0 < < 2.

Find the 2-volume of the 2-parallelogram in spanned by the vectors v₁ = (0, - 1, -2, -1) and v₂ = (1, 3, 7, 1).

Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product 11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11.

Consider the unit cube Q = in with the standard orientation given by .Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in .

The k-volume of a k-parallelogram in spanned by the k vectors, ,......, is given by det(A), where A is the k × k matrix whose columns are the components of the vectors.