MHB Proving Finite Domain Identity Element: Tips & Tricks

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To prove that every finite domain has an identity element, one should consider the properties of finite sets and the definitions of identity elements in algebraic structures. Key considerations include the structure's closure, associativity, and the existence of inverses. It is essential to apply fundamental theorems related to finite groups and ring theory, as these can provide insights into the existence of identity elements. Additionally, exploring examples of finite domains can help clarify the concept. Understanding these principles is crucial for tackling the problem effectively.
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How can I prove that every finite domain has an identity element?
How should I think about the problem and what should I take into consideration?
 
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Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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