Let's solve **Problem 1-16** step-by-step using index notation. ## Given Explicitly verify the vector identities using index notation: (a) (1.8.5)\(_{1,2,3}\) (b) (1.8.5)\(_{4,5,6,7}\) (c) (1.8.5)\(_{8,9,10}\) Assume equation (1.8.5) contains standard vector identities, commonly found in textbooks. Let's use the typical ones: ### Typical Vector Identities (in Index Notation) 1. **Divergence of a gradient:** \[ \nabla \cdot (\nabla f) = \nabla^2 f \] In index notation: \[ \partial_i \partial_i f \] 2. **Curl of a gradient:** \[ \nabla \times (\nabla f) = 0 \] In index notation: \[ (\nabla \times \nabla f)_i = \epsilon_{ijk} \partial_j \partial_k f = 0 \] (because mixed partials commute and \(\epsilon_{ijk}\) is antisymmetric) 3. **Divergence of a curl:** \[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \] In index notation: \[ \partial_i (\epsilon_{ijk} \partial_j A_k) = \epsilon_{ijk} \partial_i \partial_j A_k = 0 \] (again, due to symmetry) Let's address each part: --- ## (a) (1.8.5)\(_{1,2,3}\): Let's assume these refer to: 1. \(\nabla \cdot (\nabla f) = \nabla^2 f\) 2. \(\nabla \times (\nabla f) = 0\) 3. \(\nabla \cdot (\nabla \times \mathbf{A}) = 0\) ### 1. \(\nabla \cdot (\nabla f)\): - Index notation: \(\partial_i \partial_i f\) - This is the definition of the Laplacian: \(\nabla^2 f\). ### 2. \(\nabla \times (\nabla f)\): - Index notation: \((\nabla \times \nabla f)_i = \epsilon_{ijk} \partial_j \partial_k f\) - Since \(\partial_j \partial_k f = \partial_k \partial_j f\), and \(\epsilon_{ijk}\) is antisymmetric, their contraction is zero: \[ \epsilon_{ijk} \partial_j \partial_k f = 0 \] ### 3. \(\nabla \cdot (\nabla \times \mathbf{A})\): - Index notation: \(\partial_i (\epsilon_{ijk} \partial_j A_k) = \epsilon_{ijk} \partial_i \partial_j A_k\) - By symmetry of partial derivatives and antisymmetry of \(\epsilon_{ijk}\), this is zero. **Thus, all three are verified.** --- ## (b) (1.8.5)\(_{4,5,6,7}\): Assume these are: 4. \(\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}\) 5. \(\nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A})\) 6. Product rules, etc. Let's explicitly verify (4): ### 4. \(\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}\) - Left: \((\nabla \times (\nabla \times \mathbf{A}))_i = \epsilon_{ijk} \partial_j (\epsilon_{klm} \partial_l A_m)\) - Expand: \[ \epsilon_{ijk}\epsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl} \] - So, \[ = (\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}) \partial_j \partial_l A_m \] \[ = \delta_{il}\delta_{jm}\partial_j \partial_l A_m - \delta_{im}\delta_{jl} \partial_j \partial_l A_m \] \[ = \partial_j \partial_i A_j - \partial_j \partial_j A_i \] \[ = \partial_i (\partial_j A_j) - \partial_j \partial_j A_i \] - That is: \[ = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \] --- ## (c) (1.8.5)\(_{8,9,10}\): Likely refer to triple products and related identities. For example: 8. \(\nabla \cdot (f \mathbf{A}) = f (\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f)\) 9. \(\nabla \times (f \mathbf{A}) = (\nabla f) \times \mathbf{A} + f (\nabla \times \mathbf{A})\) 10. Product rules for divergence and curl. Let's verify (8): ### 8. \(\nabla \cdot (f \mathbf{A})\): - Index notation: \(\partial_i (f A_i) = (\partial_i f) A_i + f (\partial_i A_i)\) - Which is: \[ = \mathbf{A} \cdot (\nabla f) + f (\nabla \cdot \mathbf{A}) \] --- ## **Final Answers (Summary Table):** | Identity | Index Notation | Verification | |----------|---------------|--------------| | 1. \(\nabla \cdot (\nabla f)\) | \(\partial_i \partial_i f\) | Laplacian: \(\nabla^2 f\)| | 2. \(\nabla \times (\nabla f)\) | \(\epsilon_{ijk} \partial_j \partial_k f\) | 0 (commuting partials + antisymmetry)| | 3. \(\nabla \cdot (\nabla \times \mathbf{A})\) | \(\epsilon_{ijk} \partial_i \partial_j A_k\) | 0 | | 4. \(\nabla \times (\nabla \times \mathbf{A})\) | \(\epsilon_{ijk}\partial_j(\epsilon_{klm}\partial_l A_m)\) | \(\nabla(\nabla\cdot \mathbf{A})-\nabla^2\mathbf{A}\) | | 8. \(\nabla \cdot (f \mathbf{A})\) | \(\partial_i (f A_i)\) | \((\partial_i f)A_i + f(\partial_i A_i)\) | If you need explicit verification for more identities, let me know the exact equations (1.8.5)\(_{n}\) from your textbook, and I can verify them step-by-step!