91Ó°ÊÓ

Partial derivatives are a fundamental concept in calculus, especially in multivariable functions. They indicate how a function changes as one specific variable changes while keeping other variables constant. For instance, in the given exercise, the function \(u\) is defined in terms of two variables, \(r\) and \(t\). By computing the partial derivatives \(\frac{\partial u}{\partial t}\) and \(\frac{\partial u}{\partial r}\), we understand how \(u\) varies with each variable independently.

Here’s a summary of the partial derivatives we computed in the exercise:

• The partial derivative of \(\sin \left(\frac{r}{t}\right)\) with respect to \(t\): \( -\frac{r \cos \left(\frac{r}{t}\right)}{t^2} \)
• The partial derivative of \( \ln \left(\frac{t}{r}\right) \) with respect to \( t \): \( \frac{1}{t} \)

Combining both, we have: \( \frac{\partial u}{\partial t} = -\frac{r \cos \left(\frac{r}{t}\right)}{t^2} + \frac{1}{t} \).

Similarly for the variable \( r \):

• The partial derivative of \( \sin \left(\frac{r}{t}\right) \) with respect to \( r \): \( \frac{\cos \left(\frac{r}{t}\right)}{t} \)
• The partial derivative of \( \ln \left(\frac{t}{r}\right) \) with respect to \( r \): \( -\frac{1}{r} \)

Combining both, we have: \( \frac{\partial u}{\partial r} = \frac{\cos \left(\frac{r}{t}\right)}{t} - \frac{1}{r} \).

This step-by-step process allows us to isolate the influence of each variable, making it easier to analyze multivariable functions.

The chain rule is a powerful tool in calculus that helps us find the derivative of composite functions. In simpler terms, it allows us to differentiate functions nested within other functions. When applying the chain rule to partial derivatives, we need to take care of each layer of the function separately.

In the given exercise, the chain rule is crucial for finding derivatives for terms like \( \sin \left(\frac{r}{t}\right)\). Here’s an illustration:

• For \( \frac{\partial}{\partial t} \left( \sin \left(\frac{r}{t}\right) \right)\),
  treat \( \frac{r}{t} \) as a single entity. The outer function is the sine function, and its derivative is the cosine function. This gives us \( \cos \left(\frac{r}{t}\right) \).
• Next, handle the inner function \( \frac{r}{t} \). The derivative with respect to \( t \) is \( -\frac{r}{t^2} \).
• Combining these by the chain rule, we get: \( \cos \left(\frac{r}{t}\right) \cdot \left( -\frac{r}{t^2} \right) = -\frac{r \cos \left(\frac{r}{t}\right)}{t^2} \).

Similar steps are followed for other terms, ensuring we correctly differentiate the nested functions. Understanding and properly applying the chain rule is pivotal for tackling more complex differentiation problems.

Simplification of expressions is the process of condensing complex mathematical statements into their simplest form. This makes them easier to work with and understand, especially in multi-step problems.

Let's look at how we simplified expressions in the final step of the given exercise:

• Start by substituting the partial derivatives back into the expression \( t \frac{\partial u}{\partial t} + r \frac{\partial u}{\partial r} \).
• The substitution leads to: \[ t \cdot \left( -\frac{r \cos \left(\frac{r}{t}\right)}{t^2} + \frac{1}{t} \right) + r \cdot \left( \frac{\cos \left(\frac{r}{t}\right)}{t} - \frac{1}{r} \right) \]
.

• Distribute \( t \) and \( r \) in the respective terms: \[ t \left( -\frac{r \cos \left(\frac{r}{t}\right)}{t^2} + \frac{1}{t} \right) + r \left( \frac{\cos \left(\frac{r}{t}\right)}{t} - \frac{1}{r} \right) \]


• Simplify each term individually: \[ -\frac{r \cos \left(\frac{r}{t}\right)}{t} + 1 + \frac{r \cos \left(\frac{r}{t}\right)}{t} - 1 \]


• Notice that \( -\frac{r \cos \left(\frac{r}{t}\right)}{t} \) and \( \frac{r \cos \left(\frac{r}{t}\right)}{t} \) cancel each other out, leaving \( 1 - 1 \), which simplifies to 0.

This final verification step shows us that: \( t \frac{\partial u}{\partial t} + r \frac{\partial u}{\partial r} = 0 \). Simplification brings clarity, making complex equations more manageable and confirming our solution's accuracy.