Understanding Divergence in Vector Calculus: Definition, Examples, and Applications, Study notes of Analytical Geometry and Calculus

Download Understanding Divergence in Vector Calculus: Definition, Examples, and Applications and more Study notes Analytical Geometry and Calculus in PDF only on Docsity! Section 19.3: The Divergence of a Vector Field Question: Consider the figure below. Imagine that you were interested in trying to find a way to quantify, for lack of a better phrase, the “amount of flux emanating from a single point” on a vector field like the one above. It is clear that the vector field is radiating away from the origin, so we should imagine that the origin is a point from which a lot of flux emanates. How can you make this idea concrete? 2 Definition of Divergence GEOMETRIC DEFINITION OF DIVERGENCE: The divergence, or flux density, of a smooth vector field ~F , written div ~F , is a scalar-valued function defined by div ~F (x, y, z) = lim Volume(S)→0 ∫ S ~F · d ~A Volume(S) . Here S is a sphere centered at (x, y, z), oriented outward, that contracts down to (x, y, z) in the limit. CARTESIAN COORDINATE DEFINITION OF DIVERGENCE: If ~F = F1 ~i + F2 ~j + F3 ~k, then div ~F = ∇ · ~F = ∂F1 ∂x + ∂F2 ∂y + ∂F3 ∂z Example: Using the geometric definition of divergence, compute the divergence ~F (~r) = ~r at the origin. 5 2. For each of the following vector fields, sketched in the xy-plane, decide if the divergence is positive, negative, or zero at the indicated point. 6 3. A smooth vector field ~F has div ~F (1, 2, 3) = 5. Estimate the flux of ~F out of a small sphere of radius 0.01 centered at the point (1, 2, 3). 4. The flux of ~F out of a small sphere of radius 0.1 centered at (4, 5, 2), is 0.0125. Estimate div ~F at (4, 5, 2).