Question

# Is there any mathematical tool or other efficient way to transform from component form to vector form of partial differential equation?

In the field of solid mechanics, Navier’s partial differential equation of linear elasticity for material in vector form is:
(λ+G)∇(∇⋅f) + G∇2f = 0, where f = (u, v, w)
The corresponding component form can be evaluated by expanding the ∇ operator and organizing it as follows:
For x-component (u):
(λ+2G)*∂2u/∂x2 + G*(∂2u/∂y2 + ∂2u/∂z2) + (λ+G)*(∂2v/(∂x∂y) + ∂2v/(∂x∂z)) = 0
However, I find it difficult to convert from the component form back to its compact vector form using the combination of divergence, gradient, and Laplacian operators, especially when there are coefficients involved.
Does anyone have any experience with this? Any advice would be appreciated.

Doan Cong Dinh
Hanoi University of Science and Technology
Yi-De Liou
University of Texas at Arlington
Dear Doan Cong Dinh,
Thank you for your professional input, and I appreciate this valuable article.
I will try to grasp the concepts of quaternion analysis.

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