This version of Gauss’s Law relates a local property of the field (its divergence) to a local property of charge at that position in space (the charge per unit volume at that position in space). This is the version of Gauss’ Law that is usually seen in advanced textbooks and in Maxwell’s unified theory of electromagnetism. So, the tangent plane to the surface given by f (x,y,z) k f ( x, y, z) k at (x0,y0,z0) ( x 0, y 0, z 0) has the equation, This is a much more general form of the equation of a tangent plane than the one that we derived in the previous section. 1.2 Vector Components and Dummy Indices Let Abe a vector in R3. Where \(\rho\) is the charge per unit volume at a specific position in space. This says that the gradient vector is always orthogonal, or normal, to the surface at a point. The slopes in other directions will be called the directional derivatives.\] Temperature T is a scalar, and will certainly be a function of a position vector x (x,y,z) and may also be a function of time t: T T(x,t). Recall that the slopes due north and due west are the two partial derivatives. To give you a feeling for the issues, suppose you were interested in the temperature T of water in a river.
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