【MIT 18.952 微分形式 (Differential Forms) 】习题（有伤）通关记录 01

2023-07-08 00:08 作者:Timm31415926 0人读过 | 我要投稿

教材: MIT MATH18.952讲义 (https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf)

第一章 Multilinear Algebra

1.1 Backgrounds

章节1.1主要是复习一遍线性代数的基本知识。

1.1 习题

1.1.i Ans: Just do it be definition!

1.1.ii Ans: Just check:

$%5Ctext%7Bi)%20It%20spans%3A%20%7D%5Cforall%20v%5Cin%20%5Cmathbb%7BR%7D%5En%2C%20v%3D%5Csum_%7Bi%7Dc_ie_i.$

$%5Ctext%7Bii)%20It%20is%20linearly%20independent%3A%20%7D%0A%5Csum_%7Bi%7Dc_ie_i%3D0%5Cimplies%20c_i%3D0%2C%20%5Cforall%20i.%20$

Ans: Just check:

a) Bilinearly: $B(v_1%2Bv_2%2C%20w)%3DB(v_1%2Cw)%2BB(v_2%2Cw)%2C%20%5Cforall%20v%2C%20w%5Cin%20%5Cmathbb%7BR%7D%5En.$

b) Symmetry: $B(v%2Cw)%3DB(w%2Cv)%2C%20%20%5Cforall%20v%2C%20w%5Cin%20%5Cmathbb%7BR%7D%5En.$

c) Postivity: $B(v%2Cv)%5Cgeq%200%2C%20%5Cforall%20v%5Cin%20%5Cmathbb%7BR%7D%5En.%20(B(v%2Cv)%3D0%20%5Ciff%20v%3D0.)$

These all follows from definition of addition and multiplication of real numbers.

1.2 Quotient Spaces and Dual Spaces

章节1.2里面的内容更深一点，但也属于线性代数的范畴。

1.2 习题

1.2.i Ans:

1.2.ii Ans:

1.2.iii Ans:

1.2.iv Ans:

1.2.v Ans:

1.2.vi Ans:

1.2.vii Ans:

1.2.viii Ans:

1.2.ix Ans:

1.2.x Ans:

1.2.xi Ans:

1.2.xii Ans:

其它资参考料：

1）微分形式入门课（https://www.youtube.com/playlist?list=PL22w63XsKjqzQZtDZO_9s2HEMRJnaOTX7）：这位教授所有的视频都很推荐，讲得非常通俗易懂。

2）教科书：John Hubbard_ Barbara Burke Hubbard - Vector Calculus, Linear Algebra, and Differential Forms_ A Unified Approach-Matrix Editions (2015)

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