In a nutshell, **vector calculus** is calculus for functions that output vectors instead of numbers.
For example, the wind map from today's weather forecast pictures a function that takes in a location \(P=(x,y)\) and returns the wind velocity vector \( \vec{V}(x,y), \) which is displayed as an arrow at \( P.\)

Each entry of \( \vec{V} \) is a function of several variables, so it looks like we can just apply multivariable calculus and call it a day. Nothing could be further from the truth!

In our course, we'll uncover beautiful relationships between vector derivatives and integrals, and use them to solve important problems in partial differential equations, wave mechanics, electricity and magnetism, and much more!

This course engages you with expertly designed problems, animations, and interactive three-dimensional visualizations all designed to help you hone your vector calculus skills.

This unit, in particular, sets the stage for our first chapter, which provides a compact introduction to the essential ideas of the calculus of vectors.

The central object of our course is the

vector field, which is an assignment of a vector to every point in some subset of space.

The **gradient** \( \nabla f = \left \langle \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right \rangle \) provides an example familiar from multivariable calculus.

We learned there that \( \nabla f(x,y,z) \) is perpendicular to the level set \( f = c\) at \( (x,y,z).\) \( \nabla f\) changes from point to point, so we can think of it as a vector field.

The touch interactive visualization below shows a level set \( f = c \) of \( f = \sqrt{x^2+ y^2+z^2}\) in red together with the gradient vector field \( \nabla f \) (blue).

Zoom in and out and rotate to explore this 3D vector field, then select a true statement about \( \nabla f\) from the options provided.

If you've ever held a compass or seen the northern or southern lights, you're aware of Earth's natural magnetism.

The temperature of a cup of coffee, the flow of a river, and the amount of air in a child's balloon are probably more familiar to you from day-to-day experience.

Some are vector fields; others (the **scalars**) lack directionality and just have magnitude.

Select from the options all quantities that can also be modeled by a vector field.

To graph a vector field, we draw the arrow for \( \vec{V}(\vec{x})\) with base at \(\vec{x}\) for each point on a chosen grid, as in last problem's visualization. \( \\\\ \)

**Flow lines** (aka **field lines**) also help us see important vector field features. A flow line is a curve that “follows the arrows”; if we're on the flow line at \(\vec{x},\) \( \vec{V}(\vec{x}) \) points the way to our next stop along the curve. We include here a two-dimensional example (right) and a three-dimensional one (below).

\[ \\\\ \]

flow lines as a means of understanding one of the most important vector field derivatives: the **divergence**.

The animation on the right shows what happens to a rectangle (red) when every point in it flows along the vector field (purple).

It turns out that how much this rectangle distorts under flow is directly related to the divergence, which we turn to next.

As the name suggests, the **divergence** measures how much a vector field spreads out from a given point. We'll cover divergence in detail later in the course, but this qualitative description will do for right now.

Below are two vector fields. Select the one that has the greatest divergence from the origin.

It'll take some work with flow lines to prove this, but the divergence of a two-dimensional vector field \( \vec{V}(x,y) = \big\langle V_{x}(x,y) , V_{y}(x,y) \big\rangle \) at \( (x,y) \) is \[ \text{div}\big(\vec{V}\big) = \frac{\partial V_{x}}{\partial x} + \frac{\partial V_{y}}{\partial y}.\] Option I in the previous problem (which just circled the origin) is explicitly given by \( \langle -y, x \rangle \) and thus has divergence \[ \text{div}\big( \langle -y, x \rangle \big) = \frac{\partial }{\partial x}[-y] + \frac{\partial}{\partial y}[x]=0.\]

Compute the divergence for the vector field of option II given explicitly by \[ \vec{V}(x,y) = \left \langle \frac{x}{\sqrt{x^2+y^2}} , \frac{y}{\sqrt{x^2+y^2}} \right \rangle. \]

**Hint:** If you want to avoid partial differentiation, only one option has all the right properties for the divergence near the origin.

The formula for divergence can be derived through a careful analysis of vector field flow lines. These curves can also teach us much about another very important derivative called the **curl**.

The flow of a river is like a 2D vector field; the strength and direction of current at a point is assigned an arrow there.

If we dump a few X-shaped paddle wheels into the stream, they'll move with the flow and also rotate as fluid strikes the paddles.

Each X is assigned a vector (called the **curl**) pointing along the \(z\)-axis and providing the axis of rotation. Its length is related to the spin rate. If it points up (down), the X spins counterclockwise (clockwise).

The curl of a two-dimensional vector field is given by \[ \text{curl}\big(\vec{V}\big)(x,y) = \left[ \frac{\partial V_{y}}{\partial x} - \frac{\partial V_{x}}{\partial y} \right] \hat{k},\] as we'll see in greater detail later in the course.

Compute the curl of the vector field \( \langle -y , x \rangle \) that we encountered earlier in the unit.

Just as divergence and curl provide two different vector field derivatives, **line integrals** and **surface integrals** give us different ways of integrating with fields.

Like \( \frac{d}{dx} \) and \( \int \, dx, \) these integrals and derivatives are related. For example, in analogy to
\[ \int\limits_{[a,b]} f'(x)\, d x = f(b)-f(a),\]
we'll prove in Surface Integrals and Divergence a similar result relating the total divergence in a region \( R \) to a special sum over boundary values called a **surface integral**:
\[ \iiint\limits_{R} \text{div}\big( \vec{V}\big) \, d \vec{x} = \iint\limits_{\partial R} \vec{V}\cdot d \vec{A}.\]
As a pair, the final two units of this chapter touch on the basics of surface integrals and this relation to the divergence. Before we get there, we'll make a detour into the world of motion next.

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