As i noticed in my former [post](https://steemit.com/math/@beoped/math-computation-vector-calculus-identities-with-symbolic-computation-ep1-notation), at this post we will see the equations in detail. 

![](https://steemitimages.com/DQmej5cKpxZicbbbzKY7x8Ld23zXDEqsNgF9zxRXMsExYjN/image.png)

where A,B are vectors and f,g are scalar. 

Here we will derive these 16 equation one by one. If you are familiar with its symbol and definition of inner product and cross product it is just nothing but a  computation. 




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- 1

Using the properties of determinant which its value is invariant under the exchange of odd number of rows or columns

![](https://steemitimages.com/DQmYwPGvcjjbYgGUgJENB3dHMeKrx5F4rVwVyUQH5qjB9AC/image.png)


- 2

Using the two products of Levi-civita symbol with one repeated index

![](https://steemitimages.com/DQmWB3C8moANNQ8p2Lt9iLohcKi5LPAVDUc5EeYNMzm1Gcs/image.png)



- 3

First let me write down the form of curl and divergence

![](https://steemitimages.com/DQmdxDPGJ3tTVoSr9BvVCC3QWiVtwU3hCnshkSVjpGdZmzA/image.png)

Then automatically from symmetric argument (since partial derivatives commutes) we have 

![](https://steemitimages.com/DQmahh67FWuTym8XQowx8mV97wUsaKWzf7Zo1NhHjQYyDrU/image.png)




- 4

Note f is scalar, since gradient of scalar reproduce a vector we can write, ![](https://steemitimages.com/DQmSmTHQVSLdDdnQWTdaYAmKcLwx2hHFLHTRbV8vnMpSFdu/image.png) and again with same reason in 3

![](https://steemitimages.com/DQmSVag9iA5CHdHTwuRUjJcp6XyhpbrAc1kiFAWMpBNFkp5/image.png)






- 5

Using the two products of Levi-civita symbol with one repeated index

![](https://steemitimages.com/DQmbBCx6CxL14yyYDW9CgV3Yspb9aSsFQA6Bq6X6ZawHYVv/image.png)


- 6

Using product rule, 
![](https://steemitimages.com/DQmZoLbsp6P1yLCQt12D1VJNbeGCxBniY4rNJf34EbvYGZ3/image.png)


- 7
re-labeling the index properly and using the two products of Levi-civita symbol with one repeated index


![](https://steemitimages.com/DQmdULuYXXPHLpmjqJNKPNqUsKsp1sqirUN7aDM6erAfrmc/image.png)


- 8
using product rule 
![](https://steemitimages.com/DQmYHSK51GCwPej69NbEKeDT67GrsKHXUiezdyRUnTtSn8Q/image.png)



- 9
using product rule and the definition of cross product
![](https://steemitimages.com/DQmRvvaEizHTHTNsyqrY8emgNaEa3xYVXHy5j1LHopZBcFp/image.png)


- 10
using product rule, definition of cross product and the two products of Levi-civita symbol with one repeated index

![](https://steemitimages.com/DQmcEfUFqCC2cv2bGPjCJvq4izDNj1Kmf3QJS17BLpWmMrE/image.png)




- 11
using product rule, definition of cross product and the two products of Levi-civita symbol with one repeated index


![](https://steemitimages.com/DQmcL8mDMKCmTffXV2yg5BxoTrJs4wkmXmvWD5RX2zRBGBk/image.png)




- 12

using the definition of cross product and the two products of Levi-civita symbol with one repeated index


![](https://steemitimages.com/DQmXkh7t3UZmWK58PNeNjpChMAVdcsW8X1Lh4mgPjy2fmsm/image.png)



- 13

![](https://steemitimages.com/DQmP6T46WjRtRoqoVpqidpwD3KVc8t5Wf6JX6ByLoT51UoH/image.png)



- 14
From the determinant property which det(A) = det(A^T), 
![](https://steemitimages.com/DQmXvXA28Vmgpo4m12un9yJ8prGaRhzXUCBBxyoaUQZzTiJ/image.png)



- 15
It naturally comes from equation 13 and the properties of cross product, i.e., ![](https://steemitimages.com/DQmW24GPddSehWSDh4v7LkGdCmnkoVaBkEDwdgwiH8zQZq5/image.png)


- 16

![](https://steemitimages.com/DQmdKibSsn6tV72Niv5bUNkmQXQszzQhM1becefPxJkYkWN/image.png)
 
Now i guess you can do the other vector calculus identities via the same process