(Copyright©2023 by Daniel B. Sedory.)
Obtaining the Second Order Partial Differentials:
The second order partial derivative of the function Upon comparison of what follows to the equations for And after multiplying through to eliminate the parentheses, we get: Which after combining some of the terms, leaves us with:
Obtaining the Second Order Partial Differentials:
Again, as noted above, the second order partial derivative of the function After carrying out the differentiations, we obtain:
The Sum of All Three Second Order Partial Differentials Here we sum up all three of the previously derived terms of the second order differentials of Ten of the terms cancel out immediately. However, in order to reduce all of the remaining terms to (Equation S), trigonometric identities, such as,
Examples of how the terms above can be reduced: After applying similar algebraic operations to all the terms, the equation does in fact reduce to:
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