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The image contains two mathematical problems (ΘΕΜΑ 2 and ΘΕΜΑ 3). I will help you solve the contour integrals mentioned in the problems. Let's start by identifying and translating each problem:

### ΘΕΜΑ 2:
The problem asks to compute the contour integral:

\[
\int_{(x,y,z) \in E} \left( x^2 + y^2 - z^2 \right) \, ds
\]

where \((x, y, z) \in \mathbb{R}^3\). However, more context might be needed to identify the specific contour or surface \(E\).

### ΘΕΜΑ 3:
This problem involves proving a theorem using a parametric representation of a curve. Given the function:

\[
\gamma(t) = 2\cos t \, \mathbf{i} + 2\sin t \, \mathbf{j}
\]

for \( t \in [0, 2\pi] \), this defines a circle centered at the origin with a radius of 2. The contour integrals along this curve are to be evaluated. The problem involves proving something about the intersections and relationships between vectors.

### Let's address these contours:
1. **For ΘΕΜΑ 2**, we need to identify the specific surface or contour \(E\) to evaluate the integral correctly.
  
2. **For ΘΕΜΑ 3**, it specifies a parametric curve representing a circle. To proceed:
   - The first step is to compute \(\gamma(t)\) and understand the motion along the path.
   - Next, evaluate the integral with respect to \( \gamma(t) \).

Without more precise details about \(E\), it's challenging to solve ΘΕΜΑ 2 directly. However, if you clarify the contour or surface \(E\) for ΘΕΜΑ 2, I can proceed with detailed calculations for both problems.

Would you like me to:
1. Provide step-by-step solutions for ΘΕΜΑ 3?
2. Explain how to set up the contour integrals in detail?
3. Offer hints on common contour integral techniques?
4. Explore the geometric interpretation of the curves and surfaces involved?
5. Discuss potential applications of contour integrals in physics or engineering?

**Tip:** Contour integrals often rely on parametrization of curves and using Green's or Stokes' Theorem to simplify evaluations.

Explore detailed solutions for ΘΕΜΑ 2 and ΘΕΜΑ 3 involving contour integrals and parametric curves. Understand the application of Green's Theorem and Stokes' Theorem in evaluating these integrals. Suitable for advanced undergraduate students studying advanced calculus.

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