Let's address the problems you've listed one by one:

Question 2(b)

Problem:
We have the functions: $u = \frac{x}{\sqrt{1 - r^2}}, \quad v = \frac{y}{\sqrt{1 - r^2}}, \quad w = \frac{z}{\sqrt{1 - r^2}}, \quad \text{where} \quad r^2 = x^2 + y^2 + z^2$ and we need to show that: $\frac{\partial(u, v, w)}{\partial(x, y, z)} = \frac{-1}{\sqrt{1 - r^2}^5}$

Solution outline:

1. The Jacobian of $u, v, w$ with respect to $x, y, z$ is calculated by determining the determinant of the matrix formed by the partial derivatives of each $u, v, w$ with respect to $x, y, z$.

2. For example:

   • $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, and $\frac{\partial u}{\partial z}$ need to be computed and similarly for $v$ and $w$.
   • Once the matrix is formed, its determinant will give the Jacobian determinant.

This calculation involves differentiating each function and finding the determinant of a 3x3 matrix. We'll proceed with detailed differentiation steps if you'd like.

Question 3: Test the Convergence

We are asked to test the convergence of the series: $\sqrt{ \frac{1}{3^2 + b} } + \sqrt{ \frac{2}{3^2 + b} } + \sqrt{ \frac{3}{3^2 + b} } + \cdots$ where $a > 0$.

To test the convergence, we can apply comparison tests or ratio tests. Let's perform the test using comparison principles:

1. The general term of the series is: $a_n = \sqrt{\frac{n}{3n^2 + b}}$
2. Compare this series with a simpler form, say $\frac{1}{n^{3/2}}$, which is a p-series with $p > 1$. Based on this comparison, we can conclude if the series converges or not.

After more detailed analysis, we can determine convergence.

Question 4: Test the Convergence

We are given the series: $\sum_{n=1}^{\infty} \left( \sqrt[3]{n^3 + 1} - n \right)$

Solution Outline:

1. Simplify the general term: $a_n = \sqrt[3]{n^3 + 1} - n$ As $n \to \infty$, notice that $\sqrt[3]{n^3 + 1}$ behaves asymptotically like $n$. Therefore, the series behaves like a difference of $n$, which suggests divergence.

2. Use a series comparison or limit test to rigorously conclude whether it converges or diverges.

Question 5: Solve the Differential Equation

We are given the differential equation: $(3x^2y^4 + 2xy)dx + (2x^3y^3 - x^2)dy = 0$

Solution Outline:

1. Check if the differential equation is exact by computing the partial derivatives:

   • Compute $\frac{\partial M}{\partial y}$, where $M = 3x^2y^4 + 2xy$,
   • Compute $\frac{\partial N}{\partial x}$, where $N = 2x^3y^3 - x^2$.

2. If the equation is exact, solve for the potential function $\psi(x, y)$ such that $\frac{\partial \psi}{\partial x} = M$ and $\frac{\partial \psi}{\partial y} = N$.

Let me know if you'd like a step-by-step process for any of the questions.

Would you like detailed steps for any particular question or have specific questions?

Here are some related questions to expand your understanding:

1. How do you find the Jacobian for a transformation?
2. What are the common convergence tests used in series analysis?
3. What is an exact differential equation, and how do you solve it?
4. How can asymptotic analysis help determine series convergence?
5. What are the different methods to solve partial differential equations?

Tip: For series convergence, always check if the series simplifies to a p-series, geometric series, or apply known convergence tests like the ratio or root test.