Lec 23: Flux; normal form of Green's theorem | MIT 18.02 Multivariable Calculus, Fall 2007

The script begins with an introduction to the concept of flux in vector fields, specifically within the context of a plane. Flux is defined as a line integral involving the dot product of a vector field and a unit normal vector to a curve. The importance of the normal vector and its orientation is emphasized, with the convention that it points to the right of the curve as one travels along it. The script also highlights the significance of the formula F dot n ds, which represents the definition of flux and is crucial for understanding the concept.

This paragraph delves deeper into the meaning of flux, explaining it as a line integral that sums the dot product of the vector field and the normal vector along the curve. The process involves breaking the curve into small pieces, calculating the dot product for each piece, and then adding these values together. The analogy of work from a previous lesson is used to draw parallels with flux, with the key difference being the physical interpretation of the integrals. The paragraph also introduces the concept of flux in the context of a fluid flow, where it measures the amount of fluid passing through a curve per unit time.

The script provides a detailed explanation of flux in the context of a fluid flow, using the analogy of a river and a dam to illustrate how flux measures the amount of water passing through a given curve. It explains that flux is calculated by considering the normal component of the velocity vector field at each point along the curve and summing these components. The importance of choosing the correct direction for the normal vector is reiterated, and the concept of net flow through the curve is introduced, with positive and negative values representing flow in opposite directions.

The script presents examples of computing flux geometrically, starting with a circle of radius 'a' centered at the origin and a vector field that points radially outward. It explains how the normal vector to the circle is always perpendicular to the radius, leading to a simple calculation of flux as the integral of the constant magnitude of the vector field along the circle. The result is shown to be 2πa², indicating the amount of 'stuff' flowing out of the circle. The paragraph also contrasts this with a scenario where the vector field is tangent to the circle, resulting in zero flux, as there is no normal component to the field.

This paragraph transitions from geometric interpretations to calculating flux using coordinates. It discusses the general method for computing line integrals for flux when a geometric approach is not feasible. The script introduces the concept of expressing the normal vector in terms of its components and how to relate these components to the tangential components of the curve. The importance of the conventions used in the calculations is highlighted, and the paragraph sets the stage for a more detailed explanation of the coordinate-based calculation method.

The script provides a detailed explanation of how to calculate flux using the components of the vector field and the curve. It introduces the notation for the components of the vector field as P and Q, and explains how to express the line integral of F dot n ds in terms of these components. The paragraph demonstrates that the line integral can be transformed into an integral of -Q dx + P dy, which can then be evaluated along the curve by expressing x and y in terms of a single parameter. The process is likened to the computation of work, with the key difference being the components involved in the integral.

The script introduces Green's theorem as a tool for calculating flux along a closed curve. It presents the theorem in the context of replacing a line integral with a double integral, making the computation more manageable. The divergence of the vector field is defined, and its role in the double integral is explained. The paragraph also discusses the physical interpretation of the divergence, relating it to the expansion or contraction of a flow in a plane.

This paragraph further explores the concept of divergence, providing a physical interpretation of what it represents in the context of a vector field. Divergence is described as a measure of how much a flow is expanding or contracting, with positive divergence indicating expansion and negative divergence indicating contraction. The script uses the analogy of a gas expanding to fill more volume or a system of pumps creating more water to explain the concept. The importance of divergence in understanding the source rate or the amount of fluid being added to a system is emphasized.

The final paragraph summarizes the key points discussed in the script, focusing on the calculation of flux and the interpretation of divergence. It reiterates the method of using Green's theorem to compute flux for a closed curve and the physical significance of divergence in understanding the behavior of a vector field. The script concludes by reinforcing the importance of these concepts in the study of vector fields and their applications.