Gauss Divergence Theorem. Get the DEEPEST Intuition.

this is what the divergence integral theorem looks like in its full splendor first let's look at the right hand side of the equation the a represents a surface enclosing any volume for example the surface of a cube a sphere or the surface of any three-dimensional body you can think of the small circle around the integral indicates that this surface must satisfy a condition the surface must be closed in other words it must not contain any holes so that the quality is met mathematically the surface a is thus a closed surface the f is a vector field and represents either the electric field or the magnetic field when considering the maxwell equations so it is a vector with three components d a is an infinitesimal surface element that is an infinitely small surface element of the considered surface a as you may have already noticed the a in the d a element is shown in boldface so it is a vector with a magnitude and a direction the magnitude of the d a element indicates the area of this small piece of the surface the d a element is orthogonal to the surface and by definition points out of the surface the dot between the vector field and the d a element represents the so-called scala product the scala product is a way to multiply two vectors so here the scalar product between the vector field and the da element is formed the scalar product is defined as follows as you can see from the definition the first second and third components of the two vectors are multiplied and then added up the result of the scalar product is no longer a vector but an ordinary number a so-called scalar to understand what this number means you must first know that any vector can be written as the sum of two other vectors one vector that is parallel to the d a element let's call it f parallel and another vector that is orthogonal to the d a element let's call it f orthogonal another mathematical fact is that the scalar product of two orthogonal vectors always yields zero which means that in our case scalar product between the part f orthogonal and the d a element is zero however the scalar product between the part f parallel and the d a element is generally not zero so now you can see what the scalar product on the right hand side of the equation does it just picks out the part of the vector field that is exactly parallel to the d a element the remaining part of the vector field that points in the orthogonal direction is eliminated by the scalar product subsequently the scalar products for all locations of the considered surface a are added up that is the task of the integral the right hand side of the divergence integral theorem thus sums up all the components of the vector field f that flow into or flow out of the surface a such an integral in which small pieces of a surface are summed up is called surface integral if as in this case the integrand is a vector field this surface integral is called the flux phi of the vector field f through the surface a this description is based on what this surface integral means it measures how much of the vector field f flows out or flows into a considered surface a if the vector field f in this surface integral is an electric field e then this surface integral is called electric flux through the surface a and if the vector field f is a magnetic field b the surface integral is called magnetic flux through the surface a now let's look at the left-hand side of the theorem v is a volume but not any volume it is the volume enclosed by the surface a dv is an infinitesimal volume element in other words an infinitely small volume piece of the considered volume v the upside down triangle is called nabla operator and it has three components like a vector its components however are not numbers but derivatives corresponding to the space coordinates the first component is the derivative with respect to x the second component is the derivative with respect to y and the third component is the derivative with respect to z such an operator like the noble operator only takes effect when applied to a field and that also happens in this integral the noble operator is applied to the vector field by taking the scalar product between the double operator in the vector field as you can see it is the sum of the derivatives of the vector field with respect to the space coordinates x y and z such a scalar product between the nuble operator and a vector field f is called the divergence of the vector field f the result at the location x y z is no longer a vector but a scalar which can be either positive negative or zero if the divergence at location x y z is positive then there is a source of a vector field f at this location if this location is enclosed by surface then the flux through the surface is also positive the vector field so to speak flows out of the surface if the divergence at location x y z is negative then there is a sink of a vector field f at this location if this location is enclosed by surface then the flux through the surface is also negative the vector field flows into the surface if the divergence at location x y z disappears then that location is neither a sink nor a source of the vector field the vector field does not flow out or into or it flows in as much as out so the two amounts cancel each other out subsequently the divergence that is the sources and sinks of the vector field is summed up at each location within the volume using the integral such an integral where small pieces of volume are summed up is called volume integral so let's summarize the statement of the divergence integral theorem on the left side is the sum of the sources and sinks of the vector field within a volume and on the right side is the total flux of the vector field through the surface of that volume and the two sides should be the same the divergence integral theorem thus states that the sum of the sources and sinks of a vector field within a volume is the same as the flux of the vector field through the surface of that volume [Music]

Channel: Physics by Alexander FufaeV

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Id: zZqxbwl3Dno

Length: 7min 52sec (472 seconds)

Published: Sun Dec 19 2021