Taylor series with two variables

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The discussion focuses on the Taylor series expansion for functions with two variables, emphasizing the importance of using partial derivatives instead of total derivatives. It highlights that second-order and third-order total differentials are correctly identified, but the notation should be adjusted to reflect partial derivatives. Additionally, it points out that mixed partial derivatives, such as ∂²f/∂x∂y, are not necessarily equal to ∂²f/∂y∂x. A correction is noted regarding the omission of dxdy in the second term of the second derivative. The conversation underscores the nuances in differentiating functions of multiple variables.
JasonHathaway
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Hi everyone,

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The Attempt at a Solution



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It is correct so far.

ehild
 
So the equations of d^2f and d^2f are correct? What are they called? (Total derivatives?)
 
They are second-order and third-order total differentials. And you should write partial derivatives \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} instead df/dx and df/dy, and so on... And the partial derivative \frac{\partial^2 f}{\partial x\partial y} are not always the same as \frac{\partial^2 f}{\partial y\partial x}.

And you omitted dxdy from the second term of d^2 f

ehild
 
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