Calculus 3: Surface Integrals and Flux (Video #33) | Math with Professor V

This paragraph introduces the concept of surface integrals, building upon the understanding of integrals of single and multiple variables over different domains. The lecture focuses on integrating a function over a surface defined by z = g(x, y), and later, the integration of a vector field over a surface. The process involves partitioning the surface into pieces, evaluating the function at a point in each piece, and summing these values over the surface area, represented by ds. The integral is set up as a double integral when the surface is expressed as a function of x and y, with ds calculated using the partial derivatives of g with respect to x and y.

The paragraph presents an example of calculating a surface integral over a triangular surface in the xz-plane. It begins by defining the vertices of the triangle and deriving the plane's equation, x + y + z = 1. The integrand involves x and z, prompting the choice of the xz-plane for integration. The surface is parametrized to eliminate y from the integral, and ds is expressed in terms of dx and dz. The integral is set up with appropriate limits and simplified to a double integral over the region D, resulting in the final answer of 3/24 after integration.

This section discusses the evaluation of surface integrals on parametric surfaces, where the surface is represented by a vector-valued function r(u, v). The integral takes the form of a double integral over the domain D, with the function f(x, y, z) replaced by its parametrized form f(r(u, v)). The magnitude of the cross product of the partial derivatives with respect to u and v gives ds for the parametrically defined surface. An example is provided where the surface is a helicoid, and the integral is set up using the parametrization of the helicoid, resulting in the final answer of 4π/3 after integration.

The paragraph delves into integrating functions over more complex surfaces that require breaking down the surface into multiple parts for integration. An example is given where the surface is defined by the intersection of a cylinder, a plane y=0, and another plane x+y=2. The paragraph explains the process of sketching the solid and its bounding surfaces, and then setting up three separate surface integrals, one for each surface, to find the total integral over the solid.

This paragraph provides a detailed example of calculating the flux through a paraboloid defined by z = x^2 + y^2 - 9, bounded by a rectangle in the xy-plane. The flux is calculated by integrating the dot product of a given vector field with the normal vector to the surface. The vector field is adjusted to fit the paraboloid, and the normal vector is determined based on the surface's orientation. The resulting integral is simplified and evaluated over the specified region, yielding a final numerical value for the flux.

The paragraph explains how to calculate flux across surfaces defined parametrically, where the normal vector to the surface is found by taking the cross product of partial derivatives with respect to the parameters. The flux is then computed as the surface integral of the dot product of the vector field and the normal vector, with the integral simplified by the cancellation of the magnitude of the cross product in the ds term. An example is provided where the flux of a vector field across the surface of a solid enclosed by specific equations is calculated, resulting in a total flux of 4π.