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Relating Unit Vectors to a Jacobian Matrix
In this post, I relate coefficients of unit vectors to derivatives and to a Jacobian matrix that was used in a previous post. Unit Vectors Three unit vectors for a right-handed spherical coordinate system are $ \vec{e}_r = \sin \theta \cos\phi \vec{e}_x + \sin \theta \sin\phi \vec{e}_y + \cos \theta \vec{e}_z $ $ \vec{e}_{\theta} = \cos\theta…
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The Radial Unit Vector
In this post, I find an expression for the radial unit vector, $\vec{e}_r$. The three unit vectors in the following digram form a right-handed spherical coordinate system. For a description of this diagram, refer to the following YouTube video. The Radial Unit Vector in Terms of Spherical Coordinates Suppose $r=1$. Using vector addition, $\vec{r} =…
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The Polar Unit Vector
Consider a spherical coordinate system. Let a point be represented by $(r, \theta, \phi)$, in that order. Now that the order of the coordinates is established, I can define unit vectors that form a right-handed coordinate system. Suppose the radial unit vector $\vec{e}_r$ points radially outward from the origin to the point, and the polar…
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The Azimuthal Unit Vector
In this post, I write the azimuthal unit vector $\vec{e}_{\phi}$ in terms of Cartesian coordinates. Here, $\phi$ is the azimuthal angle in the $x-y$ plane. As noted in this post, $\vec{e}_{\phi}$ points in the direction of increasing $\phi$. Geometrical Setup Since $\vec{e}_{\phi}$ is perpendicular to the line segment from the origin to the point $(x,y,0)$, I am…