Question

Problem 2 Using index notation, demonstrate mathematically the following identities: 1- \underline{u} \times \underline{v} = -\underline{v} \times \underline{u} 2- \nabla \cdot (\nabla \underline{a}) = \nabla^2 \underline{a} 3- \nabla^2 (\underline{a}\underline{b}) = \underline{a}\nabla^2 \underline{b} + 2\nabla \underline{a} \cdot \nabla \underline{b} + \underline{b}\nabla^2 \underline{a} 4- \nabla \cdot (\underline{a}\underline{b}) = \nabla \underline{a} \cdot \underline{b} + \underline{a}\nabla \cdot \underline{b} 5- \nabla \times \phi = 0 6- \nabla \cdot (\nabla \times \underline{v}) = 0 7- (\underline{a} \times \underline{b}) \cdot (\underline{c} \times \underline{d}) = (\underline{a} \cdot \underline{c})(\underline{b} \cdot \underline{d}) - (\underline{a} \cdot \underline{d})(\underline{b} \cdot \underline{c})

          Problem 2
Using index notation, demonstrate mathematically the following identities:
1- \underline{u} \times \underline{v} = -\underline{v} \times \underline{u}
2- \nabla \cdot (\nabla \underline{a}) = \nabla^2 \underline{a}
3- \nabla^2 (\underline{a}\underline{b}) = \underline{a}\nabla^2 \underline{b} + 2\nabla \underline{a} \cdot \nabla \underline{b} + \underline{b}\nabla^2 \underline{a}
4- \nabla \cdot (\underline{a}\underline{b}) = \nabla \underline{a} \cdot \underline{b} + \underline{a}\nabla \cdot \underline{b}
5- \nabla \times \phi = 0
6- \nabla \cdot (\nabla \times \underline{v}) = 0
7- (\underline{a} \times \underline{b}) \cdot (\underline{c} \times \underline{d}) = (\underline{a} \cdot \underline{c})(\underline{b} \cdot \underline{d}) - (\underline{a} \cdot \underline{d})(\underline{b} \cdot \underline{c})

Problem 2
Using index notation, demonstrate mathematically the following identities:
1- u ×v = -v ×u
2- ∇·(∇a) = ∇^2 a
3- ∇^2 (ab) = a∇^2 b + 2∇a ·∇b + b∇^2 a
4- ∇·(ab) = ∇a ·b + a∇·b
5- ∇×ϕ= 0
6- ∇·(∇×v) = 0
7- (a ×b) ·(c ×d) = (a ·c)(b ·d) - (a ·d)(b ·c)

Added by Kathryn D.

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