https://file.aiquickdraw.com/mathbot/file/1726226950982129tffj1.jpg

The problem asks to find the value of \( r \) in Monge's equation, which is:

\[
Rr + Ss + Tt = V
\]

To determine the value of \( r \), it is helpful to recall that Monge's equation deals with a first-order partial differential equation and that the variables \( r \), \( s \), and \( t \) often correspond to the partial derivatives of a function \( z(x, y) \).

Typically:
- \( r = \frac{\partial z}{\partial x} \)
- \( s = \frac{\partial z}{\partial y} \)

The equation presented in the image involves the expression for \( r \), which is found by differentiating. The choices involve derivatives of expressions related to \( p \), \( q \), and \( s \), and seem to stem from the calculus of variations or a system of PDEs.

### Steps:
1. Identify which expression best fits for the partial derivative \( \frac{\partial z}{\partial x} \).
2. From the four options, determine the correct derivative form in terms of \( p \), \( q \), \( s \), and other variables.

### Let's examine the answer choices:
- **(a)** \( \frac{dp + s \, dy}{dx} \)
- **(b)** \( \frac{dp - s \, dy}{dx} \)
- **(c)** \( \frac{dq - s \, dx}{dy} \)
- **(d)** \( \frac{dq + s \, dx}{dy} \)

To calculate \( r \), it seems that we would look for a derivative with respect to \( x \), so we can focus on choices (a) and (b). Now, if we analyze carefully, based on known forms of Monge's equation, the derivative involving \( p \), \( dy \), and whether it is added or subtracted will determine the correct form of \( r \).

The correct answer appears to be **(a)** \( \frac{dp + s \, dy}{dx} \), as it aligns with the structure of Monge's equation where \( r \) corresponds to a combination of terms involving derivatives with respect to \( x \).

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Would you like further details on Monge's equation, or how these derivatives are computed?

Here are 5 related questions to deepen understanding:
1. How is Monge's equation derived in differential geometry?
2. What is the geometric significance of the terms \( r \), \( s \), and \( t \) in Monge's equation?
3. How do partial derivatives relate to Monge's equation in surface theory?
4. What are practical applications of Monge's equation in physics or engineering?
5. How can we verify the solution for \( r \) using a known surface \( z(x, y) \)?

**Tip:** When solving for terms in differential equations, keep track of whether the derivative is with respect to \( x \) or \( y \), as this determines the correct form of the expression.

The value of r in the Monge's equation Rr + Ss + Tt = V is ….

Find the value of r in Monge's equation Rr + Ss + Tt = V. This problem involves concepts from partial differential equations and calculus, specifically related to Monge's equation.

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