Substitution of variables in differential expressions.
Substitution of variables in differential expressions.
Often in differential expressions, derivatives with respect to certain variables need to be expressed in terms of derivatives with respect to new variables.
Examples.
1. Transform the equation
Solution.
Let's express the derivatives of
Let's substitute the found values of derivatives and the expression
Therefore,
Answer:
2. Transform the equation
Solution.
Let's express the derivatives of
Let's substitute the obtained expressions of derivatives into the given equation. We get
Thus, we have obtained the solution.
Answer:
3. Transform the equation
Solution.
We have
We substitute the expressions for
Answer:
4. To transform the equation
using the new independent variables
Solution.
We express the partial derivatives of
We have
Next, we find
Let's substitute the found expressions of derivatives into the given equation:
Answer:
5. To transform the equation
Solution.
We express the partial derivatives
We have:
Taking into account the formula
Let's substitute the found expressions for
Thus,
Answer:
Tags: Substitution of variables, calculus, derivative, differential expressions, mathematical analysis