91Ó°ÊÓ

Partial derivatives are an essential concept when dealing with functions of several variables. Unlike the regular derivative, which measures the rate of change of a function with respect to one variable, partial derivatives allow us to understand how a function changes as each individual variable changes separately.

For example, in the exercise, we have the functions:

• \( x = \frac{2u}{u^2 + v^2} \)
• \( y = -\frac{2v}{u^2 + v^2} \)

To find the Jacobian, which is a matrix of these partial derivatives, we take the derivative of each function \((x, y)\) with respect to each variable \((u, v)\).
By studying these derivatives, we can see how each variable influences a function while keeping all other variables constant. This process requires evaluating expressions involving complex fractions, often using rules like the quotient rule to simplify the steps.

The determinant is a special value that can be calculated from a square matrix. In the context of the Jacobian, the determinant helps us understand the change in volume or area as a transformation is applied by a function.

For the task of finding the Jacobian determinant \( \frac{\partial(x, y)}{\partial(u, v)} \), the determinant is calculated from the 2x2 matrix formed by the partial derivatives. In our solution, the determinant of this matrix indicates how much distortion occurs when mapping points from one coordinate system to another.
The computation involves:

• Multiplying the diagonal elements.
• Subtracting the product of the off-diagonal elements.

Mathematically, this is expressed for our specific problem as:\[\text{det}(J) = \left(\frac{2v^2 - 2u^2}{(u^2 + v^2)^2}\right)\left(\frac{-2u^2 + 2v^2}{(u^2 + v^2)^2}\right) - \left(-\frac{4uv}{(u^2 + v^2)^2}\right)^2\]which simplifies to zero, indicating a special kind of transformation.

The quotient rule is a method used in calculus to find the derivative of a ratio of two differentiable functions. It is particularly useful when you are dealing with functions like in this exercise, where both numerators and denominators are variable expressions.

The quotient rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(x) \) is given by:\[f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\]In our exercise, this rule allows us to compute complex partial derivatives efficiently. For instance, in computing \( \frac{\partial x}{\partial u} \), we found:\[\frac{(u^2 + v^2)(2) - 2u(2u)}{(u^2 + v^2)^2}\]The expression shows how the numerator's derivative is adjusted by the denominator's influence, ensuring accurate results when functions are not straightforward. Understanding this rule is vital for tackling problems where variables are interdependent.