The problem in the images you provided appears to involve rectangular parallelepipeds and unit vectors in reference frames. The questions ask about determining the partial derivatives of vector functions with respect to certain parameters and the reference frames.

From what I can gather:

Vector definitions:

• $\mathbf{u} = \mathbf{a_1} + 2\mathbf{a_2} + 3\mathbf{a_3}$
• $\mathbf{v} = \mathbf{b_1} + \mathbf{c_2} + \mathbf{d_3}$
• $\mathbf{w} = \mathbf{d_1} + 2\mathbf{d_2} + q_3 \mathbf{d_3}$

The partial derivatives are requested for the vector $\mathbf{v}$ in various reference frames and variables, with the results given as:

• $A \frac{\partial \mathbf{v}}{\partial q_1}, B \frac{\partial \mathbf{v}}{\partial q_1}, C \frac{\partial \mathbf{v}}{\partial q_2}, C \frac{\partial \mathbf{v}}{\partial q_3}, D \frac{\partial \mathbf{v}}{\partial q_2}, D \frac{\partial \mathbf{v}}{\partial q_1}$

The magnitudes provided are: $\left(1 + \cos^2(q_2)\right)^{1/2}, 0, 1, 0, 1, 0$

These represent the values for the corresponding partial derivatives in each case.

Would you like a detailed explanation of how these results were obtained or any clarification regarding the setup or partial derivatives? Additionally, here are some related questions to expand your understanding:

1. What are the geometric interpretations of the unit vectors $a_1, a_2, a_3$ in the given reference frames?
2. How do the angles $q_1, q_2, q_3$ affect the orientation of each parallelepiped?
3. What is the role of each vector ($\mathbf{u}, \mathbf{v}, \mathbf{w}$) in their respective reference frames $A, B, C, D$?
4. How does the cosine term in $(1 + \cos^2(q_2))^{1/2}$ arise in the context of rotation or orientation changes?
5. How would the partial derivative change if different vectors were used, say for vector $\mathbf{w}$?

Tip:

When working with vector functions, especially in reference frames, always pay attention to how the variables and angles affect orientation and scaling through their trigonometric relationships.