First, use [ itex ] and [ /itex ] or [ tex ] and [ itex ] (without the spaces) on this forum rather that "$". So your question isforumfann said:Could anyone help me on this question:
Suppose ${e_i}$ is an orthonormal frame, is $\nabla_{e_1}e_j$ is 0 if i is not equal to j?
Any answers or suggestion will be highly appreciated.
An orthonormal frame is a set of vectors that are mutually perpendicular and have unit length. This type of frame is commonly used in mathematics and physics to simplify calculations and define geometric properties.
An orthonormal frame is used to define the gradient of a function at a point. The gradient is a vector that points in the direction of steepest increase of the function, and its magnitude is equal to the rate of change of the function in that direction.
In an orthonormal frame, $\nabla_{e_1}e_j$ represents the directional derivative of the vector $e_j$ in the direction of the vector $e_1$. It is a measure of how $e_j$ changes as we move in the direction of $e_1$.
In an orthonormal frame, the vectors $e_i$ and $e_j$ are perpendicular to each other, meaning they have no component in the same direction. Thus, the directional derivative of $e_j$ in the direction of $e_i$ is 0, as there is no change in the direction of $e_j$ when moving in the direction of $e_i$.
Yes, an orthonormal frame can be used in any coordinate system. It is a geometric concept that is independent of the coordinate system being used. However, the specific vectors that make up the frame may be different in different coordinate systems.