# Tag Archives: vector field

## Math2111: Chapter 2: Vector fields and the operator ∇. Section 2: Divergence and Curl

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We discuss now derivatives of vector fields. To that end we introduce the del operator ∇. Throughout we assume that the vector field $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ and the scalar field $f:\mathbb{R}^3\to\mathbb{R}$ are $\mathcal{C}^1$, i.e. have at least one continuous partial derivative in each coordinate (similarly we write $\mathcal{C}^2$ if the functions are assumed to have continuous partial derivative of order at least two). Continue reading

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## Math2111: Chapter 2: Vector fields and the operator ∇. Section 1: Vector fields

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this part we deal with vector fields. These are useful for describing fluid flow or force.

Imagine a fluid moving in a pipe in a steady flow. (In physical applications the vector field may also depend on time, but we do not consider such instances here.) Then at each point we can draw a velocity vector corresponding to the velocity of the fluid at this particular point to obtain the velocity field of the fluid. If we put a particle in the fluid, then, at each point, the direction of the velocity vector is tangent to the movement of the particle and the length of the vector corresponds to the speed of the particle. See
here for a picture on fluid flow or here for some animations on fluid flow. Similarly one can draw force fields, see for instance here and here. The force field of a magnetic dipole is shown here. Continue reading