Lec 30: Line integrals in space, curl, exactness... | MIT 18.02 Multivariable Calculus, Fall 2007

The script begins with an acknowledgment of the support for MIT OpenCourseWare and a brief apology for an abrupt ending in the previous lecture. The main topic of the lecture is introduced as line integrals in 3D, which parallels the 2D case but with the additional complexity of the z-coordinate. The diffusion equation is derived from two pieces of information: the physical principle of flow from high to low concentration and the divergence theorem. The diffusion equation is expressed as ∂u/∂t = -div(F), where F is the flow vector field and u is the concentration function. The lecture then transitions into the computation of line integrals in 3D, using the dot product of the vector field with the differential displacement vector.

This paragraph delves into the specifics of calculating line integrals in 3D, emphasizing the method of parameterizing a curve and integrating with respect to that parameter. An example is provided with a vector field and a curve defined by parametric equations. The process of differentiating the parametric equations to find dx, dy, and dz, and then substituting these into the line integral formula is demonstrated. The example concludes with the computation of work done by the field along the curve, resulting in an integral that simplifies to a straightforward calculation.

The script explores the concept of conservative vector fields, where the line integral does not depend on the path taken but only on the endpoints. An example is given with a vector field and a path composed of three segments, leading from the origin to the point (1,1,1). The line integral is computed for each segment, and it is shown that even though the paths are different, the final result is the same due to the conservative nature of the field. This property is attributed to the vector field being the gradient of a potential function.

The paragraph introduces the application of the fundamental theorem of calculus to line integrals in conservative fields. It explains that once a potential function is identified, the line integral can be computed as the difference in potential values between the endpoints, rather than explicitly calculating the integral along the path. The example from the previous paragraph is revisited, and the potential function is derived, confirming the result obtained through the line integral calculation.

The criteria for a vector field to be conservative in three dimensions are discussed. It is explained that the mixed partial derivatives of the components of the vector field must be equal for the field to be conservative. This leads to three conditions that must be satisfied: ∂P/∂y = ∂Q/∂x, ∂P/∂z = ∂R/∂x, and ∂Q/∂z = ∂R/∂y. The paragraph also touches on the importance of the vector field being defined in a simply connected region for these conditions to ensure the field is conservative.

The paragraph outlines the method for finding a potential function when a vector field is conservative. It describes two approaches: one involves integrating along a chosen path from a fixed point to find the potential function at any point, and the other involves solving for the potential function using anti-derivatives based on the given components of the vector field. The process of integrating each component with respect to its corresponding variable and comparing the results to find the potential function is detailed.

The concept of curl in three-dimensional vector fields is introduced, explaining that it measures the rotation component of a field. The mathematical definition of curl is provided, and the structure of the formula is explained using the del operator and cross product notation. The geometric interpretation of curl is discussed, relating it to the angular velocity and axis of rotation in a fluid flow. The paragraph concludes with a teaser for Stokes' theorem, which will be covered in a subsequent lecture.