- #1
sunrah
- 199
- 22
Given a scalar function g defined on a manifold and a curve f:λ -> xa, the change of the function along the curve is
[itex]\frac{dg}{d\lambda} = \frac{dg}{dx^{a}}\frac{dx^{a}}{d\lambda} = T^{a}\frac{dg}{dx^{a}}[/itex]
where
[itex]\frac{dx^{a}}{d\lambda} = T^{a}[/itex] is the tangent to the curve.
The argument that I don't understand is that this
[itex]T^{a}\frac{d}{dx^{a}}[/itex]
is a vector. To me it looks like the inner product of two vectors, [itex]\vec{T} = (T_{x}, T_{y})[/itex] and ∇x,y, so looks like a scalar to me.
Also who do the coefficients of the gradient necessarily form a basis for the tangent space?
[itex]\frac{dg}{d\lambda} = \frac{dg}{dx^{a}}\frac{dx^{a}}{d\lambda} = T^{a}\frac{dg}{dx^{a}}[/itex]
where
[itex]\frac{dx^{a}}{d\lambda} = T^{a}[/itex] is the tangent to the curve.
The argument that I don't understand is that this
[itex]T^{a}\frac{d}{dx^{a}}[/itex]
is a vector. To me it looks like the inner product of two vectors, [itex]\vec{T} = (T_{x}, T_{y})[/itex] and ∇x,y, so looks like a scalar to me.
Also who do the coefficients of the gradient necessarily form a basis for the tangent space?