Flux is a particularly central scientific and mathematically idea that appears in the context of a multivariable calculus course.  Given a velocity vector field and a surface, the flux of the vector field through the surface tells us the rate at which fluid is flowing through the surface.  This leads to important ideas of balance.  If material is flowing out of a closed surface, then either the amount of material in the enclosed region is changing in time, or there is some source of material sitting within the region.

Flux it is often covered towards the end of a multivariable course, in the vector calculus portion.  In the article “Early Vector Calculus: A Path Through Multivariable Calculus” by Robert Robertson, he discusses a path through the course that makes the idea of flux more central.  To work with flux, and the related divergence theorem, students need to have been exposed to partial derivatives, vectors and the vector dot product, solid regions and their bounding surfaces, triple integrals, surface integrals, and vector fields.  For surface integrals, the cross product is useful for constructing area patches on a surface (with corresponding normal vectors) that are taken to an infinitesimal limit to construct a flux integral.

For a solid region sitting in 3-space, the divergence theorem relates the the triple integral of the divergence of a vector field over a region to the flux through the surface.  Making sense of the triple integral, being able to set it up, and being able to compute the integrand, requires familiarity with vector fields, partial derivatives, solid regions, and triple integration.  This could be done in Cartesian coordinates at first.  Making sense of the flux integral over a surface requires the notion of a surface and of an area element with a surface normal vector.  This ties into the idea of tangent planes with their normal vectors.  It also draws on knowledge of double integration.

The divergence theorem is also used over a region in 2-space with a closed curve for a boundary.  Working with it in this context requires familiarity with double integrals, vector fields, partial derivatives, and boundaries.  In addition, working with the flux across a closed curve requires familiarity with parameterizing curves, computing line integrals, and finding tangent and normal vectors to curves.

Topics often taught prior to the divergence theorem that do not obviously have immediate relevance to the theorem include directional derivatives, the chain rule, and optimization.  Further study of line integrals does make good use of background in directional derivatives and the chain rule, however the specific line integral needed for the 2D divergence theorem can perhaps be explicated without these ideas.

I often include a small amount of probability in the integration section of the course in addition to covering optimization.  It is true that these two topic areas sit somewhat to the side of other content in the course and could potentially be taught later in the course.  This is an intriguing suggestion that perhaps deserves further consideration.

Robertson, Robert L. “Early Vector Calculus: A Path Through Multivariable Calculus.” PRIMUS 23, no. 2 (2013): 133-140.