{"id":4567,"date":"2021-08-03T10:13:07","date_gmt":"2021-08-03T04:43:07","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=4567"},"modified":"2025-12-22T09:52:57","modified_gmt":"2025-12-22T04:22:57","slug":"what-is-jacobian","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-jacobian\/","title":{"rendered":"What is Jacobian?"},"content":{"rendered":"\n

Overview<\/strong>(Jacobian)<\/h2>\n\n\n\n

Jacobian is defined as- If u and v are functions of the two independent variables x and y , then the determinant,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

is known as the jacobian <\/a>of u and v with respect to x and y.<\/p>\n\n\n\n

We can write it as below<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Suppose there are three functions u, v and w of three independent variables x , y and z then,<\/p>\n\n\n\n

Suppose there are three functions u , v and w of three independent variables x , y and z then,<\/p>\n\n\n\n

The Jacobian can be defined as,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Important properties of the Jacobians<\/strong><\/h2>\n\n\n\n

Property-1-<\/strong><\/p>\n\n\n\n

If u and v are the functions of x and y , then<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Proof-  Suppose u = u(x,y) and v = v(x,y) , so that u and v are the functions of x and y,<\/p>\n\n\n\n

Now,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Interchange the rows and columns of the second determinant, we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Differentiate u = u(x,y) and v= v(x,y) partially w.r.t. u and v, we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Putting these values in eq.(1) , we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Hence proved.<\/p>\n\n\n\n

Property-2:<\/strong><\/p>\n\n\n\n

Second property is also known as chain rule.<\/p>\n\n\n\n

Suppose u and v are the functions of r and s, where r,s are the fuctions of x , y, then,\\<\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Interchange the rows and columns in second determinant<\/p>\n\n\n\n

We get,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Similarly we can prove for three variables.<\/p>\n\n\n\n

Property-3<\/strong><\/p>\n\n\n\n

If u,v,w are the functions of three independent variables x,y,z are not independent , then,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Proof: here u,v,w are independent , then f(u,v,w) = 0  \u2026\u2026\u2026\u2026\u2026\u2026.(1)<\/p>\n\n\n\n

Differentiate (1), w.r.t. x, y and z , we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n
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\"\"\/<\/figure><\/div>\n\n\n

Eliminate from 2,3,4  , we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Interchanging rows and columns , we get<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

So that<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Example: If u = x + y + z ,uv = y + z , uvw = z , then find <\/strong><\/p>\n\n\n\n

Sol.  Here we have,<\/p>\n\n\n\n

x = u \u2013 uv = u(1-v)<\/p>\n\n\n\n

y = uv \u2013 uvw = uv( 1- w)<\/p>\n\n\n\n

And  z = uvw<\/p>\n\n\n\n

So that,<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n
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\"\"\/<\/figure><\/div>\n\n
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\"\"\/<\/figure><\/div>\n\n\n

We get<\/p>\n\n\n\n

=  u\u00b2v(1-w) + u\u00b2vw<\/p>\n\n\n\n

= u\u00b2v<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n