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I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with an aspect of Kantorovitz's definition of "differential" ...

Kantorovitz's Kantorovitz's definition of "differential" reads as follows:

View attachment 7821

https://www.physicsforums.com/attachments/7822Is the "differential" as defined by Kantorovitz for real-valued functions of several real variables the same as 'the derivative" ...

If so ... is it the same situation for vector-valued functions of several real variables ...

Further to the above ... is the gradient, \(\displaystyle \bigtriangledown f\) , the differential/derivative for real-valued functions of several real variables, \(\displaystyle f\) ... ...

Help will be appreciated ...

Peter

I am currently focused on Chapter 2: Derivation ... ...

I need help with an aspect of Kantorovitz's definition of "differential" ...

Kantorovitz's Kantorovitz's definition of "differential" reads as follows:

View attachment 7821

https://www.physicsforums.com/attachments/7822Is the "differential" as defined by Kantorovitz for real-valued functions of several real variables the same as 'the derivative" ...

If so ... is it the same situation for vector-valued functions of several real variables ...

Further to the above ... is the gradient, \(\displaystyle \bigtriangledown f\) , the differential/derivative for real-valued functions of several real variables, \(\displaystyle f\) ... ...

Help will be appreciated ...

Peter

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