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### Problem Analysis

This is a problem on deformation gradient tensors and their applications in material mechanics. Here's a breakdown of the questions:

1. **Determine the deformation gradient tensor \( F \) and its inverse \( F^{-1} \):**
   - Using the provided transformation equations, you’ll need to construct the matrix \( F \) that relates the initial (reference) configuration to the deformed (spatial) configuration. This matrix can then be inverted to find \( F^{-1} \).

2. **Find the deformed vector of \( a_0 \):**
   - A material vector \( a_0 \) is given, which represents a direction or element in the undeformed (original) configuration. You’ll use \( F \) to find its deformed position, \( a \), in the new configuration by multiplying \( F \) with \( a_0 \).

3. **Using the inverse deformation gradient on vector \( b \):**
   - Here, you’ll apply the inverse deformation gradient \( F^{-1} \) to a spatial vector \( b \) to find its equivalent in the undeformed configuration. This operation allows you to backtrack from the deformed state to the original configuration.

4. **Compute the norms of \( a_0 \) and \( b \) before and after the operations:**
   - This part involves calculating the Euclidean norms (magnitudes) of the vectors before and after transformation to assess any change in their lengths. This comparison often provides insight into whether the transformation preserves length (isometric transformation) or distorts the material.

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Would you like me to proceed with the calculations for each of these steps? Let me know if you have any questions or need a specific part worked out in detail.

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### Further Questions for Deeper Understanding:

1. What does the deformation gradient tensor \( F \) physically represent in terms of material deformation?
2. Why is it useful to apply \( F^{-1} \) to vector \( b \), and what physical insights does it provide?
3. How would you interpret changes in the norms of \( a_0 \) and \( b \) after applying \( F \) and \( F^{-1} \)?
4. Are there specific conditions under which the deformation would be considered isochoric (volume-preserving)?
5. How does the deformation tensor \( F \) relate to strain measures in the material?

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### Tip:
Always verify the invertibility of \( F \) before calculating \( F^{-1} \). A non-invertible \( F \) implies a singular deformation that cannot be reversed.

Determine the deformation gradient tensor F and its inverse F^-1. Given a material line element a_0 and spatial line element b with components a_0 = [1/sqrt(2), 1/sqrt(2)] and b = [1, 0], find: (1) The deformed vector of a_0 using F. (2) The vector b in the undeformed configuration using F^-1. (3) Compute the norms of a_0 and b before and after the transformations, and comment on the results.

Explore the calculation of the deformation gradient tensor and its inverse with examples of material and spatial vectors. This problem involves finding the deformed vector of a_0, computing the vector b in the undeformed configuration using F^-1, and analyzing vector norms. Ideal for undergraduate engineering or applied mathematics students.

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