Gauss’ Theorem, also known as Gauss’ Divergence Theorem, is a theorem in vector calculus that states the total flux of a vector field through a closed surface is equal to the net divergence of the vector field within the volume enclosed by that surface. In other words, it gives an expression for computing the net flow (or flux) outwards from any region. To check whether or not this holds true for any given vector field, there are two general methods: using either Cartesian coordinates or spherical coordinates.
Checking with Cartesian Coordinates
The first method for checking Gauss’ Theorem involves using Cartesian coordinates. This consists of evaluating both sides of the equation and seeing if they match up. Specifically, on one side we have to calculate ∇ · F and on the other side we need to evaluate integrals over all faces of our closed surface S which contain outward pointing normals n̂i . First off, let’s consider what these equations look like when written out explicitly in terms of its components:
∇ · F = (∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z)
Integral Over All Faces = Σ(int_Fi n̂i· Fi dA)
Where Fi represents each face’s component along each axis i , and dA is an infinitesimal area element. Note here that this integral should be taken over all faces whose normal vectors point outwards; otherwise we wouldn’t get a valid result since those faces won’t contribute anything towards flux due to their inward pointing normal vectors cancelling each other out on opposing faces.
Now once again looking at our left-hand side equation ∇ · F , note that this can be thought simply as taking three partial derivatives along each coordinate axis respectively – one for x , y , z . So after doing this calculation separately for all three axes you’ll obtain your final answer in terms of Fx , Fy , and Fz .
Finally onto our right-hand side Integral Over All Faces equation; here we first must calculate each face’s contribution individually before summing them together over all outward facing surfaces using their respective normal vectors n̂i . For example, if calculating int_F1n̂1·F1dA then this would involve integrating across all points on surface 1 with respect to its own corresponding normal direction n̂1 which points outwards – obviously making sure only contributions from outward facing planes are being considered! Once done with these calculations combined together you’ll receive your overall resulting value as long as everything matches up perfectly between both sides then congratulations – you’ve successfully confirmed Gauss’ theorem!
Name at least two general ways how to check cartesian for Gauss’ theorem, if the normal vector point outwards.
Checking With Spherical Coordinates
The second method used to check Gauss’ Theorem involves using spherical coordinates instead of Cartesian ones. Again just like before we still need evaluate both sides separately but now instead our left hand side will consist solely of radial derivatives while right hand side will integrate across several different angles depending upon where exactly surface S lies relative itself inside 3D space – more specifically rθφ where r is radius distance away from origin point Oθ is azimuthal angle measured relative plane passing through O and φ is polar angle measured upwards from plane passing through O respectivelyHowever unlike earlier case where only three partial derivatives were needed here because spherical coordinate system has four dimensions (three spatial plus fourth temporal one!) consequently four separate radial derivatives must be computed simultaneously instead before adding them together arrive at final answer