Question 1
Let φ and ψ be two 1-forms on R
3 say φ= a
1dx
1 + a
2dx
2, ψ= b
1dx
1 + b
2dx
2. Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.
Question 2
Let φ= a
1dx
1 + a
2dx
2 + a
3 dx
3 ∈Λ
1(R
3) and ψ= b
1dx
2∧dx
3 + b
2dx
3∧dx
1+ b
3 dx
1∧dx
2 ∈Λ
2(R
3). If φ and ψ are identified with the vectors
u = a
1 i+ a
2 j+ a
3 k and
v = b
1 i+ b
2 j+ b
3 k, respectively, and if
![](data:image/png;base64, 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)
then c is equal to
◦
u × v◦
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAARACsDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD7x+FfxCi8PfBvxR4p8U6hf3NjpXibxT9oujFPfTRWsGu38aKqRq8hSOJEUKoO1EAACrx0ngz45+DPiB4itdD0TUbu41K60WHxDbx3GlXdsk1hLtEcySSxKjAl1G0HcDkEDBxzvwp8E3Wp/Bnxd4b1WG80d9W8QeLY8ywlJVguda1Bo5lVxyGjlSRTjDKykZBFc38Hf2ZfE3ww+IHhnxJqHxCt/EUOj+Eo/B72H9g/ZRJbRSb4JUYXDeXIMIHJDh8HATIwQ1m1Lb/gS/VRXzfqnKyhePxf8GP6c3zSPcZPE+jQ3jWkmrWKXSXCWjQNcoHE7pvSIrnO9k+YL1I5AxXN+LPjN4U8E+OfD3g/V7u/i8Q+II5pdMtbfSLy5W5ESlpcSxRNGCijJDMCAQccjNa7+CegXvxUh+IjzXy+KIFSCK4WVdkdsEZTbhNuCjF2YscyZPDhcKKXx3+Cv/C59F0GK08QXXhLXdD1aHVdO1ywiWSe3ZQySoobAxJFJIhzx8wJBxgxeVl6/h3/AF/DzHpd37aev9afjfoYXxg+KFt4q/ZF+JfjTwdqGp2SDwlq91p+oNa3Om3UUkdpMUlRZUjlQhlBV8DoCD0Ne1V5Z+0T4Yln/Zc+Jvh3w/pktxM/g3U9P0/TLCFpJHY2UscUMSKCWJO1VUAknAFep1rK13y7EhRRRUgFFFFABRRRQB//2Q==)
◦
u + v◦
u ∙ v◦
![](data:image/png;base64, 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)
+