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Qmechanic ♦
Qmechanic ♦
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When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$$$V^i = (d/dt)Z^i$$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.$$A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k.$$

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Christoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Christoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $$V^i = (d/dt)Z^i$$ and the components fo the acceleration are given by $$A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k.$$

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Christoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

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Slayer147
Slayer147
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When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the CristoffelChristoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Cristoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Christoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.

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Slayer147
Slayer147
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Question about the the velocity and acceleration in tensor notation

When computing the volicty of a particle moving along a curve parametrized by $Z^i(t)$ for each component i, the components of the velocity $V^i$ are given by $V^i = (d/dt)Z^i$ and the components fo the acceleration are given by $A^i=(d/dt)V^i + \Gamma^i_{jk} V^j V^k$.

My question is: why the derivative of the basis vectors doesn't appear in the expression for the velocity? Because the for the Cristoffel symbol to appear in the acceleration expression there has to be a derivative in respect to the basis vectors. What am I missing here? Any help will be appreciated.