Let R ( S , T ) = G ( U ( S , T ) , V ( S , T ) R(s, T)=G(u(s, T), V(s, T) R ( S , T ) = G ( U ( S , T ) , V ( S , T ) ], Where G G G , U U U , And V V V Are Differentiable, And The Following Applies:$[ \begin{array}{rlrl} u(3,-8) & = 5 & V(3,-8) & = -4 \ u_s(3,-8) & = -9 & V_s(3,-8) & = 4 \ u_t(3,-8) &

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Introduction

In mathematics, functions play a crucial role in describing various phenomena and relationships between different variables. The given function $R(s, t)=G(u(s, t), v(s, t))$ is a composite function, where $G$, $u$, and $v$ are differentiable functions. In this article, we will delve into the properties and behavior of this function, using the given information to derive its partial derivatives and analyze its behavior.

The Function $R(s, t)$

The function $R(s, t)$ is defined as the composition of three differentiable functions: $G$, $u$, and $v$. Specifically, we have:

$$R(s, t) = G(u(s, t), v(s, t))$$

where $G$ is a function of two variables, and $u$ and $v$ are functions of two variables as well.

Given Information

We are given the following information about the functions $u$ and $v$:

$$\begin{array}{rlrl} u(3,-8) & = 5 & v(3,-8) & = -4 \\ u_s(3,-8) & = -9 & v_s(3,-8) & = 4 \\ u_t(3,-8) & = 2 & v_t(3,-8) & = -1 \end{array}$$

This information provides us with the values of $u$ and $v$ at the point $(3, -8)$, as well as their partial derivatives with respect to $s$ and $t$.

Partial Derivatives of $R(s, t)$

To analyze the behavior of the function $R(s, t)$, we need to compute its partial derivatives with respect to $s$ and $t$. Using the chain rule, we can write:

$$\frac{\partial R}{\partial s} = \frac{\partial G}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial G}{\partial v} \frac{\partial v}{\partial s}$$

$$\frac{\partial R}{\partial t} = \frac{\partial G}{\partial u} \frac{\partial u}{\partial t} + \frac{\partial G}{\partial v} \frac{\partial v}{\partial t}$$

Computing Partial Derivatives

Using the given information, we can compute the partial derivatives of $R(s, t)$ with respect to $s$ and $t$.

First, let's compute the partial derivative of $R(s, t)$ with respect to $s$:

$$\frac{\partial R}{\partial s} = \frac{\partial G}{\partial u} (-9) + \frac{\partial G}{\partial v} (4)$$

We are not given the values of $\frac{\partial G}{\partial u}$ and $\frac{\partial G}{\partial v}$, so we cannot compute the exact value of $\frac{\partial R}{\partial s}$. However, we can write an expression for it in terms of these partial derivatives.

Next, let's compute the partial derivative of $R(s, t)$ with respect to $t$:

$$\frac{\partial R}{\partial t} = \frac{\partial G}{\partial u} (2) + \frac{\partial G}{\partial v} (-1)$$

Again, we are not given the values of $\frac{\partial G}{\partial u}$ and $\frac{\partial G}{\partial v}$, so we cannot compute the exact value of $\frac{\partial R}{\partial t}$. However, we can write an expression for it in terms of these partial derivatives.

Analyzing the Behavior of $R(s, t)$

To analyze the behavior of the function $R(s, t)$, we need to examine its partial derivatives with respect to $s$ and $t$. From the expressions we derived earlier, we can see that the partial derivatives of $R(s, t)$ depend on the partial derivatives of $G$ with respect to $u$ and $v$.

Since we are not given the values of $\frac{\partial G}{\partial u}$ and $\frac{\partial G}{\partial v}$, we cannot determine the exact behavior of $R(s, t)$. However, we can make some general observations about its behavior.

First, we can see that the partial derivative of $R(s, t)$ with respect to $s$ is a linear combination of the partial derivatives of $G$ with respect to $u$ and $v$. This means that the behavior of $R(s, t)$ with respect to $s$ is influenced by the behavior of $G$ with respect to $u$ and $v$.

Second, we can see that the partial derivative of $R(s, t)$ with respect to $t$ is also a linear combination of the partial derivatives of $G$ with respect to $u$ and $v$. This means that the behavior of $R(s, t)$ with respect to $t$ is also influenced by the behavior of $G$ with respect to $u$ and $v$.

Conclusion

In this article, we analyzed the function $R(s, t)=G(u(s, t), v(s, t))$, where $G$, $u$, and $v$ are differentiable functions. We used the given information to derive the partial derivatives of $R(s, t)$ with respect to $s$ and $t$, and examined their behavior. Our analysis showed that the behavior of $R(s, t)$ is influenced by the behavior of $G$ with respect to $u$ and $v$.

References

• [1] Calculus, by Michael Spivak
• [2] Multivariable Calculus, by James Stewart

Further Reading

• Partial Derivatives, by Wolfram MathWorld
• Chain Rule, by Wolfram MathWorld
  

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Q: What is the function $R(s, t)$?

A: The function $R(s, t)$ is a composite function, defined as $R(s, t) = G(u(s, t), v(s, t))$, where $G$, $u$, and $v$ are differentiable functions.

Q: What is the given information about the functions $u$ and $v$?

A: We are given the following information about the functions $u$ and $v$:

$$\begin{array}{rlrl} u(3,-8) & = 5 & v(3,-8) & = -4 \\ u_s(3,-8) & = -9 & v_s(3,-8) & = 4 \\ u_t(3,-8) & = 2 & v_t(3,-8) & = -1 \end{array}$$

Q: How do we compute the partial derivatives of $R(s, t)$?

A: We can compute the partial derivatives of $R(s, t)$ using the chain rule. Specifically, we have:

$$\frac{\partial R}{\partial s} = \frac{\partial G}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial G}{\partial v} \frac{\partial v}{\partial s}$$

$$\frac{\partial R}{\partial t} = \frac{\partial G}{\partial u} \frac{\partial u}{\partial t} + \frac{\partial G}{\partial v} \frac{\partial v}{\partial t}$$

Q: Can we determine the exact behavior of $R(s, t)$?

A: No, we cannot determine the exact behavior of $R(s, t)$ without knowing the values of $\frac{\partial G}{\partial u}$ and $\frac{\partial G}{\partial v}$. However, we can make some general observations about its behavior.

Q: How is the behavior of $R(s, t)$ influenced by the behavior of $G$?

A: The behavior of $R(s, t)$ is influenced by the behavior of $G$ with respect to $u$ and $v$. Specifically, the partial derivatives of $R(s, t)$ depend on the partial derivatives of $G$ with respect to $u$ and $v$.

Q: What are some general observations about the behavior of $R(s, t)$?

A: Some general observations about the behavior of $R(s, t)$ include:

• The partial derivative of $R(s, t)$ with respect to $s$ is a linear combination of the partial derivatives of $G$ with respect to $u$ and $v$.
• The partial derivative of $R(s, t)$ with respect to $t$ is also a linear combination of the partial derivatives of $G$ with respect to $u$ and $v$.

Q: What are some further reading resources for learning more about partial derivatives and the chain rule?

A: Some further reading resources for learning more about partial derivatives and the chain rule include:

• Partial Derivatives, by Wolfram MathWorld
• Chain Rule, by Wolfram MathWorld
• Calculus, by Michael Spivak
• Multivariable Calculus, by James Stewart

Q: What are some references for learning more about the function $R(s, t)$?

A: Some references for learning more about the function $R(s, t)$ include:

• Calculus, by Michael Spivak
• Multivariable Calculus, by James Stewart

Conclusion

In this article, we answered some frequently asked questions about the function $R(s, t)=G(u(s, t), v(s, t))$. We covered topics such as the definition of the function, the given information about the functions $u$ and $v$, and the computation of the partial derivatives of $R(s, t)$. We also made some general observations about the behavior of $R(s, t)$ and provided some further reading resources for learning more about partial derivatives and the chain rule.