16: Vector Calculus

# 16.1 Vector Fields #

• A vector field in .$\mathbb{R}^3$ is a function .$\vec F$ on region .$E \in \mathbb{R}^3$ that assigns each point .$(x,y,z)$ a vector .$F(x,y,z)$
• .$\vec F$ is made up of component function: .$\vec F(x,y,z) = \langle P(x,y,z)\hat i, Q(x,y,z) \hat j, R(x,y,z) \hat k\rangle$
• .$\vec F$ is continuos iff its component vectors are continuos
• .$\vec F$ is conservative (path taken doesn’t change work) iff potential function .$f(x,y,z)$ is a partial of .$\vec F$ $$\vec F = \nabla f$$
• Notice that the gradient lines are always perpendicular to the level sets
• If the function .$f$ is differentiable, .$\nabla f$ at a point is either zero or perpendicular to the level set of .$f$ at that point.
• That is, that the gradient of a function is called a gradient field which is always conservative (the fundamental theorem of calculus for line integrals)
• Conversely, a (continuous) conservative vector field is always the gradient of a function

# 16.2 Line Integrals #

• We know that the distance (length) normally is .$L = \int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2}\ dt$
• Over a vector field, we can think of the function being the density of the line (or height of particle). Therefore, we say .$ds = \int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2}\ dt$ and can write $$\int_C f(x,y) ds = \int_a^bf(x(t), y(t)) \cdot \sqrt{\bigg(\frac{dx}{dt}\bigg)^2 + \bigg(\frac{dy}{dt}\bigg)^2} dt$$ and for 3D in a slightly different form: $$\int_a^b f (\vec r (t) ) \vert \vec r'(t) \vert \Longrightarrow \int_a^b f(x(t), y(t), z(t)) \cdot \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} dt$$
• We can write .$\vec a \to \vec b$ as .$(1-t)\vec a + t\vec b$ with .$t\in[0,1]$
• Just like usual, we can break up un-integrable smooth curves, i.e $$\int_a^z f (x,y) = \int_a^b f_1(x,y) + \int_b^c f_2(x,y) + \dots \int_{\dots}^z f_n(x,y)$$

## wrt variable #

Opposed to the line integrals on .$f$ along .$C$ with respect to .$x$ both and .$y$, we can write line integral with respect to arc length as follows:

$$\int_C f(x,y) dx = \int_a^b f(x(t), y(t)) \cdot y'(t) dt$$ $$\int_C f(x,y) dy = \int_a^b f(x(t), y(t)) \cdot x'(t) dt$$

It frequently happens that line integrals with respect to .$x$ and .$y$ occur together which we abbr as

$$\int_C P(x,y)\ dx + \int_C Q(x,y)\ dy = \int_C P(x,y)\ dx + Q(x,y)\ dy$$

## Orientation #

• When we parameterize a curve, we give it a direction
• Positive: Enclosed region .$D$ is always on the left as we traverse curve .$C$ (counter-clockwise)
• Negative: Enclosed region .$D$ is always on the right as we traverse curve .$C$ (clockwise)
• The orientation represents the direction of the line
• The positive direction corresponding to increasing values of the parameter .$t$
• Doesn’t matter for regular line integrals: .$\int_C f(x,y) ds = \int_C f(x,y) ds$
• Deals with distance, .$ds$, which doesn’t depend on direction
• Does matter for field line integrals: .$\int_C f(x,y) dx \neq \int_C f(x,y) dy$
• Deals with displacement, .$dx/dy$, which depends on direction

Let .$\vec F$ be a continuous vector field defined on a smooth curve .$C$ given by a vector function .$\vec r(t), t\in[a,b]$. Then the line integral of .$\vec F$ along .$C$ is $$W = \int_C \vec F \cdot d\vec r = \int_a^b \vec F ( \vec r (t) ) \cdot (\vec r (t))'\ dt = \int_C \vec F \cdot \vec T\ ds$$

• .$\vec T(x,y,z)$ is the unit tangent vector at the point .$(x,y,z)$ on .$C$
• .$\vec F \cdot \vec T = \vec F(x,y,z) \cdot \vec T(x,y,z)$
• This equation says that work is the line integral with respect to arc length of the tangential component of the force.
• Then, for a non-conservative force i.e .$F = \langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle$ $$W = \int_a^b P(\vec r(t)) \cdot x'(t) + Q(\vec r(t)) \cdot y'(t) + R(\vec r(t)) \cdot z'(t)$$ $$\Longrightarrow \int_C P\ dx + Q\ dy + R\ dz$$
• If we flip the curve…
• …and integrate with respect to just .$x$ or .$y$ then the value flips: $$\int_{-C}f(x,y)\ dx = - \int_C f(x,y)\ dx$$
• Since .$\Delta x$ and .$\Delta y$ change sign when we reverse the orientation of .$C$.
• …and integrate with respect to arc length, the value of the line integral does not change when we reverse the orientation of the curve: $$\int_{-C}f(x,y)\ ds = \int_C f(x,y)\ ds$$
• This is because .$\Delta s$ is always positive.

# 16.3 Fundamental Thm for Line Integrals #

Let .$C$ be a smooth curve given by the vector function .$\vec r (t), t\in[a,b]$. Let .$f$ be a differentiable function of two or three variables whose gradient vector .$\nabla f$ is continuous on .$C$. Then $$\int_C \nabla f \cdot d\vec r =\int_C \vec F \cdot d\vec r = f(\vec r(b) ) f(\vec r(a))$$

## Independence of Path #

• Suppose .$C_1$ and .$C_2$ are two piecewise-smooth curves (which are called paths) that have the same initial point .$A$ and terminal point .$B$.
• Therefore, .$\int_{C_1} \nabla f \cdot d\vec r = \int_{C_2} \nabla f \cdot d\vec r$ whenever .$\nabla f$ is continuous
• In other words, line integrals of conservative vector fields are independent of path (they only depend on the start and end points)

### Plane Curves #

.$\int_C \vec F \cdot d\vec r$ is independent of path in .$D$ iff .$\int_C \vec F \cdot d\vec r = 0$ for every closed path .$C$ in .$D$
• Closed: A curve with the same end and start points: .$\vec r(b) = \vec r(a)$
• That is, only vector fields that are independent of path are conservative.

### Space Curves #

Suppose .$\vec F$ is a vector field that is continuous on an open connected region .$D$. If .$\int_C \vec F \cdot d\vec r$ is independent of path in .$D$, then .$\vec F$ is a conservative vector field on .$D$; that is, there exists a function .$f$ such that .$\nabla f = \vec F$.
• Open: For every point .$P$ in .$D$ there is a disk with center .$P$ that lies entirely in .$D$. (So .$D$ doesn’t contain any of its boundary points.)
• Connected: Any two points in .$D$ can be joined by a path that lies in .$D$.



• If .$\vec F (x,y) = P(x,y) \hat i + Q(x,y) \hat j$ is a conservative vector field, where .$P$ and .$Q$ have continuous first-order partial derivatives on a domain .$D$, then throughout .$D$ we have $$\frac{\delta P}{\delta y} = \frac{\delta Q}{\delta x}$$
• The converse of the theorem above is true on only simple curves: curves that don’t intersect itself anywhere between its endpoints
• A simply-connected region in the plane is a connected region .$D$ such that every simple closed curve in .$D$ encloses only points that are in .$D$
• Intuitively speaking, a simply-connected region contains no hole and can’t consist of two separate pieces.
.$\vec F = \langle P, Q \rangle$ is a conservative vector field on an open simply-connected region .$D$ iff both .$P$ and .$Q$ have continuous first-order partial derivatives and $$\frac{\delta P}{\delta y} = \frac{\delta Q}{\delta x} \text{ throughout } D$$

# 16.4 Green’s Theorem #

Green’s Theorem: Let .$C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let .$D$ be the region bounded by .$C$. If .$P$ and .$Q$ have continuous partial derivatives on an open region that contains .$D$, then $$\int_C \vec F \cdot d\vec r = \oint_C P\ dx + Q\ dy = \iint_D \bigg(\frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y}\bigg) dA$$

• .$dA = dx\ dy = r \cdot dr\ d\theta = \dots$
• .$\vec F(x,y) = \langle P(x,y), Q(x,y)\rangle$
• .$\oint$ implies an integral over a closed curve
• Proof is too much for me to write out here, the book does a good job though
• One important takeaway is the the shape doesn’t have to be “nice” – we can break up any shape into parts that are either Type I or II. Even though we will have overlapping lines, they cancel one another out (leaving only the boundaries) because of orientation
• Green’s Theorem should be regarded as the counterpart of the Fundamental Theorem of Calculus for double integrals
• Recall the Fundamental Theorem of Calculus is .$\int_a^b F'(x)\ dx = F(b) - F(a)$
• In both cases there is an integral involving derivatives (.$F', \delta Q/\delta x, \delta P/\delta y$) on the left side of the equation.
• And in both cases the right side involves the values of the original functions (.$F, Q, P$) only on the boundary of the domain.
• (In the one-dimensional case, the domain is an interval .$[a,b]$ whose boundary consists of just two points, .$a$ and .$b$.)

## Application: Finding Area #

• Since the area of .$D$ is .$\iint_D 1 dA$, we wish to choose .$P$ and .$Q$ so that $$\frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y} = 1$$
• Some examples of .$P/Q$ combos are

$$P(x,y)=0$$ $$Q(x,y)=x$$ $$A = \oint_C x\ dy$$

$$P(x,y)=-y$$ $$Q(x,y)=0$$ $$A = -\oint_C y\ dx$$

$$P(x,y)=-y/2$$ $$Q(x,y)=x/2$$ $$A = \frac{1}{2} \oint_C x\ dy - y\ dx$$

• Planimeters (a measuring instrument used to determine the area of an arbitrary two-dimensional shape) is an example of an application of Green Theorem
• We can also use Green’s to prove our last equation found in 16.3: $$\oint_C \vec F \cdot d \vec r = \oint_C P\ dx + Q\ dy = \iint_R \bigg( \frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y}\bigg)\ dA = \iint_R 0\ dA = 0$$

A curve that is not simple crosses itself at one or more points and can be broken up into a number of simple curves. We have shown that the line integrals of .$\vec F$ around these simple curves are all .$0$ and, adding these integrals, we see that .$\int_C \vec F \cdot d\vec r = 0$ for any closed curve .$C$. Therefore .$\int_C \vec F \cdot d\vec r$ is independent of path in .$D$. It follows that .$F$ is a conservative vector field.

# 16.5 Curl and Divergence #

• Recall that .$\nabla = \langle \frac{\delta}{\delta x} \frac{\delta}{\delta y} \frac{\delta}{\delta z} \rangle$
• 3b1b video going over this section

## Curl #

• Vector of the rotation caused by the field at a given point $$\nabla \times \vec F(x,y,z) = \begin{bmatrix}\hat i & \hat j & \hat k\\ \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta z}\\ P & Q & R\end{bmatrix} = \dots$$
• If .$\vec F$ is conservative then .$\text{curl($\vec F$) = 0}$
• We know if .$\vec F$ is conservative then .$\vec F = \nabla f$ for some .$f(x,y,z)$
• Crossing .$\nabla F$ with .$\nabla$ we get .$\langle \frac{\delta^2 f}{\delta y \delta z} - \frac{\delta^2 f}{\delta z \delta y}, \dots \rangle = \langle 0, 0,0 \rangle$

## Divergence #

• If .$\vec F (x,y,z)$ is the velocity of a fluid (or gas), then .$\text{div}(\vec F (x,y,z))$ represents the net rate of change (wrt time) of the mass of fluid (or gas) flowing from the point .$(x,y,z)$ per unit volume.
• In other words, .$\text{div}(\vec F (x,y,z))$ measures the tendency of the fluid to diverge from the point .$(x,y,z)$.
• If .$\text{div}(\vec F (x,y,z)) = 0$, then .$F$ is said to be incompressible.
• Scalar of the amount of “flow” at a given point – how much does the field expand/contract at a given point? $$\nabla \cdot \vec F(x,y,z) = \langle \frac{\delta}{\delta x} \frac{\delta}{\delta y} \frac{\delta}{\delta z} \rangle \cdot \langle P, Q, R \rangle$$

• Fun fact: .$\text{(div(curl($\vec F$)))} = \nabla \cdot (\nabla \times \vec F)= 0$
• We can use this fact to find if there exists a vector field .$\vec G$ such that .$\text{curl($\vec G$)} = \vec H$ because .$\text{div(curl($\vec G$))} = \text{div($\vec H$)} = 0$

## Vector Form of Green’s #

$$\oint_C \vec F \cdot d \vec r= \iint_D \text{(curl (\vec F))} \cdot \hat k\ dA = \bigg( \frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y} \bigg) \hat k$$

• This equation expresses the line integral of the tangential component of .$\vec F$ along .$C$ as the double integral of the vertical component of .$\text{curl($\vec F$)}$ over the region .$D$ enclosed by .$C$.
• We can write this using the normal component of .$\vec F$ too:
• With .$\vec r(t) = \langle x(t), y(t) \rangle; t \in [a,b]$
• Recall the unit tangent vector: .$\vec T(t) = \frac{1}{\vert \vec r ' (t) \vert} \langle x'(t), y'(t) \rangle$
• The outward unit normal vector to .$C$ is given by .$\vec n (t) = \frac{1}{\vert \vec r ' (t) \vert} \langle y'(t), -x'(t) \rangle$
• We can then evaluate $$\oint_C \vec F \cdot \vec n\ ds = \int_a^b (\vec F \cdot \vec n)(t) \vert \vec r'(t) \vert\ dt = \iint_D \bigg( \frac{\delta P}{\delta x} - \frac{\delta Q}{\delta y} \bigg)\ dA = \iint_D \text{div \vec F (x,y)}\ dA$$
• This says that the line integral of the normal component of .$\vec F$ along .$C$ is equal to the double integral of the divergence of .$\vec F$ over the region .$D$ enclosed by .$C$.

# 16.6 Parametric Surfaces and Their Area #

## Parametric Surfaces #

• Just like how we can describe curves with single parameter (variable) function .$\vec r(t)$, we can describe surfaces with a vector function .$\vec r(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle$
• .$\vec r(u,v)$ is called a vector-valued function defined on a region .$D$ in the .$uv$-plane
• .$x,y,z$ are the component functions of .$\vec r$, each of which have domain .$D$
• In general, a surface given as the graph of a function of .$x$ and .$y$, that is, with an equation of the form .$z=f(x,y)$, can always be regarded as a parametric surface by taking .$x$ and .$y$ as parameters and writing the parametric equations as .$x=x; y=y; z=f(x,y)$
• E.x. the plane with point .$(x_0, y_0, z_0)$ and vectors .$\langle a,b,c \rangle$ .$\langle \alpha,\beta,\gamma \rangle$ is .$\vec r(u,v) = \langle x_0, y_0, z_0 \rangle + u\langle a,b,c \rangle + v\langle \alpha,\beta,\gamma \rangle$ or .$0 = (\langle x - x_0, y - y_0, z - z_0 \rangle) \cdot (\langle a,b,c \rangle \times \langle \alpha,\beta,\gamma \rangle)$