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I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ...

I am focused on Section 3.2 The Cauchy Riemann Equations ...

I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its proof reads as follows:View attachment 9350In the above proof by Mathews and Howell we read the following:

" ... ... The partial derivatives \(\displaystyle u_x\) and \(\displaystyle u_y\) exist, so the mean value theorem for real functions of two variables implies that a value \(\displaystyle x*\) exists between \(\displaystyle x_0\) and \(\displaystyle x_0 + \Delta x\) such that we can write the first term in brackets on the right side of equation (3-17) as

\(\displaystyle u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0 + \Delta y) = u_x(x*, y_0 + \Delta y) \Delta x \) ... ... "

Can someone please explain how exactly the mean value theorem for real functions of two variables implies that a value \(\displaystyle x*\) exists between \(\displaystyle x_0\) and \(\displaystyle x_0 + \Delta x\) such that we can write the first term in brackets on the right side of equation (3-17) as

\(\displaystyle u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0 + \Delta y) = u_x(x*, y_0 + \Delta y) \Delta x\) ... ...

Peter[ NOTE ... ... In Wendell Fleming's book: "Functions of Several Variables" (Second Edition) the Mean Value Theorem reads as follows:View attachment 9351... ... ]

I am focused on Section 3.2 The Cauchy Riemann Equations ...

I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its proof reads as follows:View attachment 9350In the above proof by Mathews and Howell we read the following:

" ... ... The partial derivatives \(\displaystyle u_x\) and \(\displaystyle u_y\) exist, so the mean value theorem for real functions of two variables implies that a value \(\displaystyle x*\) exists between \(\displaystyle x_0\) and \(\displaystyle x_0 + \Delta x\) such that we can write the first term in brackets on the right side of equation (3-17) as

\(\displaystyle u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0 + \Delta y) = u_x(x*, y_0 + \Delta y) \Delta x \) ... ... "

Can someone please explain how exactly the mean value theorem for real functions of two variables implies that a value \(\displaystyle x*\) exists between \(\displaystyle x_0\) and \(\displaystyle x_0 + \Delta x\) such that we can write the first term in brackets on the right side of equation (3-17) as

\(\displaystyle u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0 + \Delta y) = u_x(x*, y_0 + \Delta y) \Delta x\) ... ...

Peter[ NOTE ... ... In Wendell Fleming's book: "Functions of Several Variables" (Second Edition) the Mean Value Theorem reads as follows:View attachment 9351... ... ]

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