- #1
merrypark3
- 30
- 0
Hello.
How can I show the Divergence of a vector field is a scalar field(in [tex]E^{3}[/tex]) ?
Should I show that Div is invariant under rotation?
[tex]x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x})[/tex]
then
[tex]\frac{\partial V^{'}_{i}(\stackrel{\rightarrow}{x})}{\partial x^{'i}}=\frac{\partial(a_{ij}V_{j} (\stackrel{\rightarrow}{x})) }{\partial(a^{ij} x^{j} ) } = \frac{\partial V_{i} (\stackrel{\rightarrow}{x})}{\partial x^{i}} [/tex]
How can I prove the last equality?
How can I show the Divergence of a vector field is a scalar field(in [tex]E^{3}[/tex]) ?
Should I show that Div is invariant under rotation?
[tex]x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x})[/tex]
then
[tex]\frac{\partial V^{'}_{i}(\stackrel{\rightarrow}{x})}{\partial x^{'i}}=\frac{\partial(a_{ij}V_{j} (\stackrel{\rightarrow}{x})) }{\partial(a^{ij} x^{j} ) } = \frac{\partial V_{i} (\stackrel{\rightarrow}{x})}{\partial x^{i}} [/tex]
How can I prove the last equality?