Spherical coordinates are of course the most intimidating for the untrained eye. But derivatives of the unit vectors are not zero all the time. 46 0 obj <>/Filter/FlateDecode/ID[<973913FD09855F8FBD5EA6C853C4F40C><003244497A267E4FA4CBBDFCA6682BED>]/Index[24 40]/Info 23 0 R/Length 102/Prev 54785/Root 25 0 R/Size 64/Type/XRef/W[1 2 1]>>stream The natural way of defining the velocity or acceleration is based on a Lagrangian description because it is the easiest (and that's how we've done it in dynamics in high school)! Using the vector identity mentioned in the preliminaries, this equation can be expanded as: $ \nabla\cdot\vec{u} = \left(\nabla u_r\right)\cdot\hat{e}_r + u_r\left(\nabla\cdot\hat{e}_r\right) + \left(\nabla u_{\theta}\right)\cdot\hat{e}_{\theta} + u_{\theta}\left(\nabla\cdot\hat{e}_{\theta}\right) + \left(\nabla u_z\right)\cdot\hat{e}_z + u_z\left(\nabla\cdot\hat{e}_z\right) $. In Cartesian coordinates, gradient, divergence, and curl are defined as below, where $ n $ is the number of spatial dimensions involved. ��x �z H�T�k���ĭ� �sa

Be careful. But, since the divergence operator is the same for all coordinate systems, we can use its implementation in Cartesian coordinates just as well as the one in cylindrical coordinates. Remembering some of the formulas from dynamics, we have, upon substitution of Eq.

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While if the field lines are sourcing in or contracting at a point then there is a negative divergence.

But Cylindrical and Spherical unit vector are location dependent unit vectors and their derivatives may be zero or may not be zero. The terms involving gradients of the components of the vector field simplify to the partial derivatives of components with respect to their corresponding directions, multiplied by the coefficients found in the previous section: $ \nabla\cdot\vec{u} = \frac{\partial u_r}{\partial r} + \frac{1}{r}\frac{\partial u_{\theta}}{\partial \theta} + \frac{\partial u_z}{\partial z} + u_r\left(\nabla\cdot\hat{e}_r\right) + u_{\theta}\left(\nabla\cdot\hat{e}_{\theta}\right) + u_z\left(\nabla\cdot\hat{e}_z\right) $. The divergence formula in cartesian coordinate system can be derived from the basic definition of the divergence. But as if now, let us just use their results as follows: So let us consider the complete terms together, The highlighted terms can be rewritten for consistency as follows. φ is called as the azimuthal angle which is angle made by the half-plane containing the required point with the positive X-axis. One could arrive at the correct formula for the gradient by performing many changes of variables, and repeat the process for the other vector derivatives. Those coefficients are not necessarily obvious, and deriving them is usually tedious if not difficult. %PDF-1.5 %���� You may think them as the constants and their derivatives be zero.

The entries of the square matrix come from the coordinate transformation itself: $ x = r \cos \theta \rightarrow \frac{\partial x}{\partial r} = \cos \theta \text{ , } \frac{\partial x}{\partial \theta} = -r\sin \theta $, $ y = r \sin \theta \rightarrow \frac{\partial y}{\partial r} = \sin \theta \text{ , } \frac{\partial y}{\partial \theta} = r\cos \theta $, $ z = z \rightarrow \frac{\partial x}{\partial z} = \frac{\partial y}{\partial z} = 0 \text{ , } \frac{\partial z}{\partial z} = 1 $, $ \begin{bmatrix} \frac{\partial \phi}{\partial r} \\ \frac{\partial \phi}{\partial \theta} \\ \frac{\partial \phi}{\partial z}\end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta & 0 \\ -r \sin\theta & r \cos\theta & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix} \frac{\partial \phi}{\partial x} \\ \frac{\partial \phi}{\partial y} \\ \frac{\partial \phi}{\partial z} \end{bmatrix} $. Derivatives of the unit vectors in different coordinate systems. In other words neither the magnitude nor the direction of the aρ is changing w.r.t. Put another way, if you imagine the radial unit vectors as the velocity of some fluid, then an infinitesimal region at each point has a greater volume of fluid leaving it than entering it. Then, and the material derivative is written as (with the capital D symbol to distinguish it from the total and partial derivatives), or, by making use of the definition of velocities ($u \equiv \frac{{\rm d}r}{{\rm}t}$, $\frac{v}{r} \equiv \frac{{\rm d}\theta}{{\rm d}t}$, and $w \equiv \frac{{\rm d} z}{{\rm d} t}$), we have, Special attention must be made in evaluating the time derivative in Eq. When we switch to the Eulerian reference, the velocity becomes a function of position, which, implicitly, is a function of time as well as viewed from the Eulerian reference. But this is not true all the time. Standard procedure for finding the Electric Field due to distributed charge. I am going to cover the derivatives of the unit vectors in the independent article.

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