Lengths and distances

If we have a vector in ${\displaystyle \mathbb {R} ^{2}}$ we can calculate its length using the Pythagorean theorem. For instance, the length of the vector ${\displaystyle (2,3)}$ is

${\displaystyle {\sqrt {3^{2}+2^{2}}}={\sqrt {13}}}$

We can generalize this to ${\displaystyle \mathbb {R} ^{n}}$ . We define a vector's length, written ${\displaystyle \|{\vec {x}}\|}$ , as the square root of the sum of the squares of each of its components. That is, if we have a vector ${\displaystyle {\vec {x}}=(x_{1},\ldots ,x_{n})}$ ,

${\displaystyle \|{\vec {x}}\|={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}}$

Now that we have established some concept of length, we can establish the distance between two vectors. We define this distance to be the length of the two vectors' difference. We write this distance ${\displaystyle d({\vec {x}},{\vec {y}})}$ , and it is

${\displaystyle d({\vec {x}},{\vec {y}})={\big \|}{\vec {x}}-{\vec {y}}{\big \|}={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}$

This distance function is sometimes referred to as a metric. Other metrics arise in different circumstances. The metric we have just defined is known as the Euclidean metric.

Open and closed balls

In ${\displaystyle \mathbb {R} }$ , we have the concept of an interval, in that we choose a certain number of other points about some central point. For example, the interval ${\displaystyle [-1,1]}$ is centered about the point 0, and includes points to the left and right of 0.

In ${\displaystyle \mathbb {R} ^{2}}$ and up, the idea is a little more difficult to carry on. For ${\displaystyle \mathbb {R} ^{2}}$ , we need to consider points to the left, right, above, and below a certain point. This may be fine, but for ${\displaystyle \mathbb {R} ^{3}}$ we need to include points in more directions.

We generalize the idea of the interval by considering all the points that are a given, fixed distance from a certain point - now we know how to calculate distances in ${\displaystyle \mathbb {R} ^{n}}$ , we can make our generalization as follows, by introducing the concept of an open ball and a closed ball respectively, which are analogous to the open and closed interval respectively.

an open ball

${\displaystyle B({\vec {a}},r)}$

is a set in the form ${\displaystyle {\Big \{}{\vec {x}}\in \mathbb {R} ^{n}{\Big |}d({\vec {x}},{\vec {a}})<r{\Big \}}}$

a closed ball

${\displaystyle {\overline {B}}({\vec {a}},r)}$

is a set in the form ${\displaystyle {\Big \{}{\vec {x}}\in \mathbb {R} ^{n}{\Big |}d({\vec {x}},{\vec {a}})\leq r{\Big \}}}$

In ${\displaystyle \mathbb {R} }$ , we have seen that the open ball is simply an open interval centered about the point ${\displaystyle x=a}$ . In ${\displaystyle \mathbb {R} ^{2}}$ this is a circle with no boundary, and in ${\displaystyle \mathbb {R} ^{3}}$ it is a sphere with no outer surface. (What would the closed ball be?)

Boundary points

If we have some area, say a field, then the common sense notion of the boundary is the points 'next to' both the inside and outside of the field. For a set, ${\displaystyle S}$ , we can define this rigorously by saying the boundary of the set contains all those points such that we can find points both inside and outside the set. We call the set of such points ${\displaystyle \partial S}$ .

Typically, when it exists the dimension of ${\displaystyle \partial S}$ is one lower than the dimension of ${\displaystyle S}$ . e.g. the boundary of a volume is a surface and the boundary of a surface is a curve.

This isn't always true; but it is true of all the sets we will be using.

A set ${\displaystyle S}$ is bounded if there is some positive number such that we can encompass this set by a closed ball about ${\displaystyle {\vec {0}}}$ . --> if every point in it is within a finite distance of the origin, i.e there exists some ${\displaystyle r>0}$ such that ${\displaystyle {\vec {x}}}$ is in S implies ${\displaystyle {\vec {x}}<r}$ .

Collision and intersection points

Say we have two different curves. It may be important to consider

• points the two curves share - where they intersect
• intersections which occur for the same value of ${\displaystyle t}$ - where they collide.

Continuity & differentiability

If we have a parametrization ${\displaystyle {\vec {f}}:\mathbb {R} \to \mathbb {R} ^{n}}$ , which is built up out of component functions in the form ${\displaystyle {\vec {f}}(t)={\big (}f_{1}(t),\ldots ,f_{n}(t){\big )}}$ , ${\displaystyle {\vec {f}}}$ is continuous if and only if each component function is also.

In this case the derivative of ${\displaystyle {\vec {f}}(t)}$ is

${\displaystyle a_{i}={\big (}f_{1}'(t),\ldots ,f_{n}'(t){\big )}}$ . This is actually a specific consequence of a more general fact we will see later.

Tangent vectors

Recall in single-variable calculus that on a curve, at a certain point, we can draw a line that is tangent to that curve at exactly at that point. This line is called a tangent. In the several variable case, we can do something similar.

We can expect the tangent vector to depend on ${\displaystyle {\vec {f}}'(t)}$ and we know that a line is its own tangent, so looking at a parametrised line will show us precisely how to define the tangent vector for a curve.

An arbitrary line is ${\displaystyle {\vec {f}}(t)=t{\vec {a}}+{\vec {b}}}$ , with: ${\displaystyle f_{i}(t)=ta_{i}t+b_{i}}$ , so

${\displaystyle f_{i}'(t)=a_{i}}$ and

${\displaystyle f'(t)={\vec {a}}}$ , which is the direction of the line, its tangent vector.

Similarly, for any curve, the tangent vector is ${\displaystyle f'(t)}$ .

Angle between curves

We can then formulate the concept of the angle between two curves by considering the angle between the two tangent vectors. If two curves, parametrized by ${\displaystyle {\vec {f_{1}}}}$ and ${\displaystyle {\vec {f_{2}}}}$ intersect at some point, which means that

${\displaystyle {\vec {f_{1}}}(s)={\vec {f_{2}}}(t)=c}$

the angle between these two curves at ${\displaystyle c}$ is the angle between the tangent vectors ${\displaystyle {\vec {f_{1}'}}(s)}$ and ${\displaystyle {\vec {f_{2}'}}(t)}$ is given by

${\displaystyle \arccos \left({\frac {{\vec {f_{1}'}}(s)\cdot {\vec {f_{2}'}}(t)}{{\Big \|}{\vec {f_{1}'}}(s){\Big \|}{\Big \|}{\vec {f_{2}'}}(t){\Big \|}}}\right)}$

Different parameterizations

One such parametrization of a curve is not necessarily unique. Curves can have several different parametrizations. For example, we already saw that the unit circle can be parametrized by g(t) = (cos at, sin at) such that t ∈ [0, 2π/a).

Generally, if f is one parametrization of a curve, and g is another, with

f(t₀) = g(s₀)

there is a function u(t) such that u(t₀)=s₀, and g'(u(t)) = f(t) near t₀.

This means, in a sense, the function u(t) "speeds up" the curve, but keeps the curve's shape.

Surfaces

A surface in space can be described by the image of a function f : R² → Rⁿ. f is said to be the parametrization of that surface.

For example, consider the function

f(α, β) = α(2,1,3)+β(-1,2,0)

This describes an infinite plane in R³. If we restrict α and β to some domain, we get a parallelogram-shaped surface in R³.

Surfaces can also be described explicitly, as the graph of a function z = f(x, y) which has a standard parametrization as f(x,y)=(x, y, f(x,y)), or implictly, in the form f(x, y, z)=c.

Rules

Since this is an almost identical formulation of limits in the single variable case, many of the limit rules in the one variable case are the same as in the multivariate case.

For f and g, mapping R^m to Rⁿ, and h(x) a scalar function mapping R^m to R, with

• f(x) → b as x → a
• g(x) → c as x → a
• h(x) → H as x → a

then:

• ${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} +\mathbf {g} )=\mathbf {b} +\mathbf {c} }$
• ${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }(h\mathbf {f} )=H\mathbf {b} }$

and consequently

• ${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {b} \cdot \mathbf {c} }$
• ${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \times \mathbf {g} )=\mathbf {b} \times \mathbf {c} }$

when H≠0

• ${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }({\mathbf {f} \over h})={\mathbf {b} \over H}}$

Continuity

Again, we can use a similar definition to the one variable case to formulate a definition of continuity for multiple variables.

If f : R^m → Rⁿ, f is continuous at a point a in R^m if f(a) is defined and

${\displaystyle \lim _{\mathbf {x} \rightarrow \mathbf {a} }\mathbf {f} (\mathbf {x} )=\mathbf {f} (\mathbf {a} )}$

Just as for functions of one dimension, if f, g are both continuous at p, f+g, λf (for a scalar λ), f·g, and f×g are continuous also. If φ : R^m → R is continus at p, φf, f/φ are too if φ is never zero.

From these facts we also have that if A is some matrix which is n×m in size, with x in R^m, a function f(x)=A x is continuous in that the function can be expanded in the form x₁a₁+...+x_ma_m, which can be easily verified from the points above.

If f : R^m → Rⁿ which is in the form f(x) = (f₁(x),...,f_n(x) is continuous if and only if each of its component functions are a polynomial or rational function, whenever they are defined.

Finally, if f is continuous at p, g is continuous at f(p), g(f(x)) is continuous at p.

Special note about limits

It is important to note that we can approach a point in more than one direction, and thus, the direction that we approach that point counts in our evaluation of the limit. It may be the case that a limit may exist moving in one direction, but not in another.

Properties

There are a number of important properties of this formulation of the derivative.

Rules of taking Jacobians

If f : R^m → Rⁿ, and h(x) : R^m → R are differentiable at 'p':

• ${\displaystyle J_{\mathbf {p} }(\mathbf {f} +\mathbf {g} )=J_{\mathbf {p} }\mathbf {f} +J_{\mathbf {p} }\mathbf {g} }$
• ${\displaystyle J_{\mathbf {p} }(h\mathbf {f} )=hJ_{\mathbf {p} }\mathbf {f} +\mathbf {f} (\mathbf {p} )J_{\mathbf {p} }h}$
• ${\displaystyle J_{\mathbf {p} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {g} ^{T}J_{\mathbf {p} }\mathbf {f} +\mathbf {f} ^{T}J_{\mathbf {p} }\mathbf {g} }$

Important: make sure the order is right - matrix multiplication is not commutative!

Alternate notations

For simplicity, we will often use various standard abbreviations, so we can write most of the formulae on one line. This can make it easier to see the important details.

We can abbreviate partial differentials with a subscript, e.g.,

${\displaystyle \partial _{x}h(x,y)={\frac {\partial h}{\partial x}}\quad \partial _{x}\partial _{y}h=\partial _{y}\partial _{x}h}$

When we are using a subscript this way we will generally use the Heaviside D rather than ∂,

${\displaystyle D_{x}h(x,y)={\frac {\partial h}{\partial x}}\quad D_{x}D_{y}h=D_{y}D_{x}h}$

Mostly, to make the formulae even more compact, we will put the subscript on the function itself.

${\displaystyle D_{x}h=h_{x}\quad h_{xy}=h_{yx}}$

If we are using subscripts to label the axes, x₁, x₂ …, then, rather than having two layers of subscripts, we will use the number as the subscript.

${\displaystyle h_{1}=D_{1}h=\partial _{1}h=\partial _{x_{1}}h={\frac {\partial h}{\partial x_{1}}}}$

We can also use subscripts for the components of a vector function, u=(u_x, u_y, u_y) or u=(u₁,u₂…u_n)

If we are using subscripts for both the components of a vector and for partial derivatives we will separate them with a comma.

${\displaystyle u_{x,y}={\frac {\partial u_{x}}{\partial y}}}$

The most widely used notation is h_x. Both h₁ and ∂₁h are also quite widely used whenever the axes are numbered. The notation ∂_xh is used least frequently.

We will use whichever notation best suits the equation we are working with.

Directional derivatives

Normally, a partial derivative of a function with respect to one of its variables, say, x_j, takes the derivative of that "slice" of that function parallel to the x_j'th axis.

More precisely, we can think of cutting a function f(x₁,...,x_n) in space along the x_j'th axis, with keeping everything but the x_j variable constant.

From the definition, we have the partial derivative at a point p of the function along this slice as

${\displaystyle {\partial \mathbf {f} \over \partial x_{j}}=\lim _{t\rightarrow 0}{\mathbf {f} (\mathbf {p} +t\mathbf {e} _{j})-\mathbf {f} (\mathbf {p} ) \over t}}$

provided this limit exists.

Instead of the basis vector, which corresponds to taking the derivative along that axis, we can pick a vector in any direction (which we usually take as being a unit vector), and we take the directional derivative of a function as

${\displaystyle {\partial \mathbf {f} \over \partial \mathbf {d} }=\lim _{t\rightarrow 0}{\mathbf {f} (\mathbf {p} +t\mathbf {d} )-\mathbf {f} (\mathbf {p} ) \over t}}$

where d is the direction vector.

If we want to calculate directional derivatives, calculating them from the limit definition is rather painful, but, we have the following: if f : Rⁿ → R is differentiable at a point p, |p|=1,

${\displaystyle {\partial \mathbf {f} \over \partial \mathbf {d} }=D_{\mathbf {p} }\mathbf {f} (\mathbf {d} )}$

There is a closely related formulation which we'll look at in the next section.

Gradient vectors

The partial derivatives of a scalar tell us how much it changes if we move along one of the axes. What if we move in a different direction?

We'll call the scalar f, and consider what happens if we move an infintesimal direction dr=(dx,dy,dz), using the chain rule.

${\displaystyle \mathbf {df} =dx{\frac {\partial f}{\partial x}}+dy{\frac {\partial f}{\partial y}}+dz{\frac {\partial f}{\partial z}}}$

This is the dot product of dr with a vector whose components are the partial derivatives of f, called the gradient of f

${\displaystyle \operatorname {grad} \mathbf {f} =\nabla \mathbf {f} =\left({\frac {\partial \mathbf {f} (\mathbf {p} )}{\partial x_{1}}},\cdots ,{\frac {\partial \mathbf {f} (\mathbf {p} )}{\partial x_{n}}}\right)}$

We can form directional derivatives at a point p, in the direction d then by taking the dot product of the gradient with d

${\displaystyle {\partial \mathbf {f} (\mathbf {p} ) \over \partial \mathbf {d} }=\mathbf {d} \cdot \nabla \mathbf {f} (\mathbf {p} )}$.

Notice that grad f looks like a vector multiplied by a scalar. This particular combination of partial derivatives is commonplace, so we abbreviate it to

${\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)}$

We can write the action of taking the gradient vector by writing this as an operator. Recall that in the one-variable case we can write d/dx for the action of taking the derivative with respect to x. This case is similar, but ∇ acts like a vector.

We can also write the action of taking the gradient vector as:

${\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},\cdots {\frac {\partial }{\partial x_{n}}}\right)}$

Divergence

If the vector function u maps Rⁿ to itself, then we can take the dot product of u and ∇. This dot product is called the divergence.

${\displaystyle \operatorname {div} \,\mathbf {u} =\nabla \cdot \mathbf {u} ={\frac {\partial u_{1}}{\partial x_{1}}}+{\frac {\partial u_{2}}{\partial x_{2}}}+\cdots {\frac {\partial u_{n}}{\partial x_{n}}}}$

If we look at a vector function like v=(1+x²,xy) we can see that to the left of the origin all the v vectors are converging towards the origin, but on the right they are diverging away from it.

Div u tells us how much u is converging or diverging. It is positive when the vector is diverging from some point, and negative when the vector is converging on that point.

Example:

For v=(1+x², xy), div v=3x, which is positive to the right of the origin, where v is diverging, and negative to the left of the origin, where v is converging.

Like grad, div is linear.

${\displaystyle \nabla \cdot \left(a\mathbf {u} +b\mathbf {v} \right)=a\nabla \cdot \mathbf {u} +b\nabla \cdot \mathbf {v} }$

Later in this chapter we will see how the divergence of a vector function can be integrated to tell us more about the behaviour of that function.

To find the divergence we took the dot product of ∇ and a vector with ∇ on the left. If we reverse the order we get

${\displaystyle \mathbf {u} \cdot \nabla =u_{x}D_{x}+u_{y}D_{y}+u_{z}D_{z}}$

To see what this means consider i·∇ This is D_x, the partial differential in the i direction. Similarly, u·∇ is the partial differential in the u direction, multiplied by |u|

Curl

If u is a three-dimensional vector function on R³ then we can take its cross product with ∇. This cross product is called the curl.

${\displaystyle \operatorname {curl} \,\mathbf {u} =\nabla \times \mathbf {u} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\D_{x}&D_{y}&D_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}$

Curl u tells us if the vector u is rotating round a point. The direction of curl u is the axis of rotation.

We can treat vectors in two dimensions as a special case of three dimensions, with u_z=0 and D_zu=0. We can then extend the definition of curl u to two-dimensional vectors

${\displaystyle \operatorname {curl} \,\mathbf {u} =D_{y}u_{x}-D_{x}u_{y}}$

This two dimensional curl is a scalar. In four, or more, dimensions there is no vector equivalent to the curl.

Example:
Consider u=(-y, x). These vectors are tangent to circles centred on the origin, so appear to be rotating around it anticlockwise.

${\displaystyle \operatorname {curl} \,\mathbf {u} =D_{y}(-y)-D_{x}x=-2}$

Example
Consider u=(-y, x-z, y), which is similar to the previous example.

${\displaystyle \operatorname {curl} \,\mathbf {u} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\D_{x}&D_{y}&D_{z}\\-y&x-z&y\end{vmatrix}}=2\mathbf {i} +2\mathbf {k} }$

This u is rotating round the axis i+k

Later in this chapter we will see how the curl of a vector function can be integrated to tell us more about the behaviour of that function.

Product and chain rules

Just as with ordinary differentiation, there are product rules for grad, div and curl.

• If g is a scalar and v is a vector, then

the divergence of gv is

${\displaystyle \nabla \cdot (g\mathbf {v} )=g\nabla \cdot \mathbf {v} +(\mathbf {v} \cdot \nabla )g}$

the curl of gv is

${\displaystyle \nabla \times (g\mathbf {v} )=g(\nabla \times \mathbf {v} )+(\nabla g)\times \mathbf {v} }$

• If u and v are both vectors then

the gradient of their dot product is

${\displaystyle \nabla (\mathbf {u} \cdot \mathbf {v} )=\mathbf {u} \times (\nabla \times \mathbf {v} )+\mathbf {v} \times (\nabla \times \mathbf {u} )+(\mathbf {u} \cdot \nabla )\mathbf {v} +(\mathbf {v} \cdot \nabla )\mathbf {u} }$

the divergence of their cross product is

${\displaystyle \nabla \cdot (\mathbf {u} \times \mathbf {v} )=\mathbf {v} \cdot (\nabla \times \mathbf {u} )-\mathbf {u} \cdot (\nabla \times \mathbf {v} )}$

the curl of their cross product is

${\displaystyle \nabla \times (\mathbf {u} \times \mathbf {v} )=(\mathbf {v} \cdot \nabla )\mathbf {u} -(\mathbf {u} \cdot \nabla )\mathbf {v} +\mathbf {u} (\nabla \cdot \mathbf {v} )-\mathbf {v} (\nabla \cdot \mathbf {u} )}$

We can also write chain rules. In the general case, when both functions are vectors and the composition is defined, we can use the Jacobian defined earlier.

${\displaystyle \left.\nabla \mathbf {u} (\mathbf {v} )\right|_{\mathbf {r} }=\mathbf {J} _{\mathbf {v} }\left.\nabla \mathbf {v} \right|_{\mathbf {r} }}$

where J_u is the Jacobian of u at the point v.

Normally J is a matrix but if either the range or the domain of u is R¹ then it becomes a vector. In these special cases we can compactly write the chain rule using only vector notation.

• If g is a scalar function of a vector and h is a scalar function of g then

${\displaystyle \nabla h(g)={\frac {dh}{dg}}\nabla g}$

• If g is a scalar function of a vector then

${\displaystyle \nabla =(\nabla g){\frac {d}{dg}}}$

This substitution can be made in any of the equations containing ∇

Second order differentials

We can also consider dot and cross products of ∇ with itself, whenever they can be defined. Once we know how to simplify products of two ∇'s we'll know out to simplify products with three or more.

The divergence of the gradient of a scalar f is

${\displaystyle \nabla ^{2}f(x_{1},x_{2},\ldots x_{n})={\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\ldots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}}$

This combination of derivatives is the Laplacian of f. It is commmonplace in physics and multidimensional calculus because of its simplicity and symmetry.

We can also take the Laplacian of a vector,

${\displaystyle \nabla ^{2}\mathbf {u} (x_{1},x_{2},\ldots x_{n})={\frac {\partial ^{2}\mathbf {u} }{\partial x_{1}^{2}}}+{\frac {\partial ^{2}\mathbf {u} }{\partial x_{2}^{2}}}+\ldots +{\frac {\partial ^{2}\mathbf {u} }{\partial x_{n}^{2}}}}$

The Laplacian of a vector is not the same as the divergence of its gradient

${\displaystyle \nabla (\nabla \cdot \mathbf {u} )-\nabla ^{2}\mathbf {u} =\nabla \times (\nabla \times \mathbf {u} )}$

Both the curl of the gradient and the divergence of the curl are always zero.

${\displaystyle \nabla \times \nabla f=0\quad \nabla \cdot (\nabla \times \mathbf {u} )=0}$

This pair of rules will prove useful.

Riemann sums

When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areas of these rectangles as their widths get smaller and smaller. For the multiple-variable case, we need to do something similar, but the problem arises how to split up R², or R³, for instance.

To do this, we extend the concept of the interval, and consider what we call a n-interval. An n-interval is a set of points in some rectangular region with sides of some fixed width in each dimension, that is, a set in the form {x∈Rⁿ|a_i ≤ x_i ≤ b_i with i = 0,...,n}, and its area/size/volume (which we simply call its measure to avoid confusion) is the product of the lengths of all its sides.

So, an n-interval in R² could be some rectangular partition of the plane, such as {(x,y) | x ∈ [0,1] and y ∈ [0, 2]|}. Its measure is 2.

If we are to consider the Riemann sum now in terms of sub-n-intervals of a region Ω, it is

${\displaystyle \sum _{i;S_{i}\subset \Omega }f(x_{i}^{*})m(S_{i})}$

where m(S_i) is the measure of the division of Ω into k sub-n-intervals S_i, and x^*_i is a point in S_i. The index is important - we only perform the sum where S_i falls completely within Ω - any S_i that is not completely contained in Ω we ignore.

As we take the limit as k goes to infinity, that is, we divide up Ω into finer and finer sub-n-intervals, and this sum is the same no matter how we divide up Ω, we get the integral of f over Ω which we write

${\displaystyle \int _{\Omega }f}$

For two dimensions, we may write

${\displaystyle \int \int _{\Omega }f}$

and likewise for n dimensions.

Iterated integrals

Thankfully, we need not always work with Riemann sums every time we want to calculate an integral in more than one variable. There are some results that make life a bit easier for us.

For R², if we have some region bounded between two functions of the other variable (so two functions in the form f(x) = y, or f(y) = x), between a constant boundary (so, between x = a and x =b or y = a and y = b), we have

${\displaystyle \int _{a}^{b}\,\int _{f(x)}^{g(x)}h(x,y)\,dydx}$

An important theorem (called Fubini's theorem) assures us that this integral is the same as

${\displaystyle \int \int _{\Omega }f}$,

if f is continuous on the domain of integration.

Order of integration

In some cases the first integral of the entire iterated integral is difficult or impossible to solve, therefore, it can be to our advantage to change the order of integration.

${\displaystyle \int _{a}^{b}\,\int _{f(x)}^{g(x)}h(x,y)\,dxdy}$

${\displaystyle \int _{c}^{d}\,\int _{e(y)}^{f(y)}h(x,y)\,dydx}$

As of the writing of this, there is no set method to change an order of integration from dxdy to dydx or some other variable. Although, it is possible to change the order of integration in an x and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integration from the graph of the limits of integration.

In higher order integration that can't be graphed, the process can be very tedious. For example, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then to dydzdx and then to dzdydx (as 3-dimensional cases can be graphed, this method would lack parsimony).

Parametric integrals

If we have a vector function, u, of a scalar parameter, s, we can integrate with respect to s simply by integrating each component of u separately.

${\displaystyle \mathbf {v} (s)=\int \mathbf {u} (s)\,ds\Rightarrow v_{i}(s)=\int u_{i}(s)\,ds}$

Similarly, if u is given a function of vector of parameters, s, lying in Rⁿ, integration with respect to the parameters reduces to a multiple integral of each component.

Line integrals

In one dimension, saying we are integrating from a to b uniquely specifies the integral.

In higher dimensions, saying we are integrating from a to b is not sufficient. In general, we must also specify the path taken between a and b.

We can then write the integrand as a function of the arclength along the curve, and integrate by components.

E.g., given a scalar function h(r) we write

${\displaystyle \int _{C}h(\mathbf {r} )\,d\mathbf {r} =\int _{C}h(\mathbf {r} ){\frac {d\mathbf {r} }{ds}}\,ds=\int _{C}h(\mathbf {r} (s))\mathbf {t} (s)\,ds}$

where C is the curve being integrated along, and t is the unit vector tangent to the curve.

There are some particularly natural ways to integrate a vector function, u, along a curve,

${\displaystyle \int _{C}\mathbf {u} \,ds\quad \int _{C}\mathbf {u} \cdot d\mathbf {r} \quad \int _{C}\mathbf {u} \times d\mathbf {r} \quad \int _{C}\mathbf {u} \cdot \,\mathbf {n} ds}$

where the third possibility only applies in 3 dimensions.

Again, these integrals can all be written as integrals with respect to the arclength, s.

${\displaystyle \int _{C}\mathbf {u} \cdot d\mathbf {r} =\int _{C}\mathbf {u} \cdot \mathbf {t} \,ds\quad \int _{C}\mathbf {u} \times d\mathbf {r} =\int _{C}\mathbf {u} \times \mathbf {t} \,ds\quad }$

If the curve is planar and u a vector lying in the same plane, the second integral can be usefully rewritten. Say,

${\displaystyle \mathbf {u} =u_{t}\mathbf {t} +u_{n}\mathbf {n} +u_{b}\mathbf {b} }$

where t, n, and b are the tangent, normal, and binormal vectors uniquely defined by the curve.

Then

${\displaystyle \mathbf {u} \times \mathbf {t} =-\mathbf {b} u_{n}+\mathbf {n} u_{b}}$

For the 2-d curves specified b is the constant unit vector normal to their plane, and u_b is always zero.

Therefore, for such curves,

${\displaystyle \int _{C}\mathbf {u} \times d\mathbf {r} =\int _{C}\mathbf {u} \cdot \,\mathbf {n} ds}$

Green's Theorem

Let C be a piecewise smooth, simple closed curve that bounds a region S on the Cartesian plane. If two function M(x,y) and N(x,y) are continuous and their partial derivatives are continuous, then

${\displaystyle \int \int _{S}({\frac {\partial N}{\partial x}}-{\frac {\partial M}{\partial y}})dA=\oint _{C}Mdx+Ndy=\oint _{C}\mathbf {F} \cdot d\mathbf {r} }$

In order for Green's theorem to work there must be no singularities in the vector field within the boundaries of the curve.

A demonstration of how a large loop can be decomposed into a family of infinitesimal loops.

Green's theorem works by summing the circulation in each infinitesimal segment of area enclosed within the curve.

An infinitesimal rectangular loop.

To show how an infinitesimal loop satisfies Green's theorem, consider the infinitesimal rectangle ${\displaystyle R=[x_{l},x_{u}]\times [y_{l},y_{u}]}$. Let ${\displaystyle (x_{c},y_{c})\in R}$ be an arbitrary point inside the rectangle, and let ${\displaystyle \partial R}$ be the counterclockwise boundary of ${\displaystyle R}$.

The circulation around ${\displaystyle \partial R}$ is approximately (the error vanishes as ${\displaystyle \Delta x,\Delta y\rightarrow 0^{+}}$):

${\displaystyle \oint _{\partial R}Mdx+Ndy\approx N(x_{u},y_{c})\Delta y-M(x_{c},y_{u})\Delta x-N(x_{l},y_{c})\Delta y+M(x_{c},y_{l})\Delta x}$ ${\displaystyle =\left({\frac {N(x_{u},y_{c})-N(x_{l},y_{c})}{\Delta x}}-{\frac {M(x_{c},y_{u})-M(x_{c},y_{l})}{\Delta y}}\right)\Delta x\Delta y}$ ${\displaystyle \approx \left({\frac {\partial N}{\partial x}}{\bigg |}_{(x_{c},y_{c})}-{\frac {\partial M}{\partial y}}{\bigg |}_{(x_{c},y_{c})}\right)\Delta x\Delta y}$ ${\displaystyle \approx \iint _{R}\left({\frac {\partial N}{\partial x}}-{\frac {\partial M}{\partial y}}\right)dxdy}$

As ${\displaystyle \Delta x,\Delta y\rightarrow 0^{+}}$, the relative errors present in the approximations vanish, and therefore, for an infinitesimal rectangle, ${\displaystyle \oint _{\partial R}Mdx+Ndy=\iint _{R}\left({\frac {\partial N}{\partial x}}-{\frac {\partial M}{\partial y}}\right)dxdy}$

Adding together a family of infinitesimal loops demonstrates Green's theorem for a single large loop.

Surface and Volume Integrals

Just as with curves, it is possible to parameterise surfaces then integrate over those parameters without regard to geometry of the surface.

That is, to integrate a scalar function V over a surface A parameterised by r and s we calculate

${\displaystyle \int _{A}V(x,y,z)\,dS=\int \int _{A}V(r,s)\det J\,drds}$

where J is the Jacobian of the transformation to the parameters.

To integrate a vector this way, we integrate each component separately.

However, in three dimensions, every surface has an associated normal vector n, which can be used in integration. We write dS=ndS.

For a scalar function, V, and a vector function, v, this gives us the integrals

${\displaystyle \int _{A}V\,\mathbf {dS} \quad \int _{A}\mathbf {v} \cdot \mathbf {dS} \quad \int _{A}\mathbf {v} \times \mathbf {dS} }$

These integrals can be reduced to parametric integrals but, written this way, it is clear that they reflect more of the geometry of the surface.

When working in three dimensions, dV is a scalar, so there is only one option for integrals over volumes.

Surface Vectors

Consider an infinitesimal surface element with an area of ${\displaystyle |dS|}$ and a unit outwards normal vector of ${\displaystyle \mathbf {n} }$. The infinitesimal "surface vector" ${\displaystyle \mathbf {dS} =\mathbf {n} |dS|}$ describes the infinitesimal surface element in a manner similar to how the infinitesimal displacement ${\displaystyle d\mathbf {r} }$ describes an infinitesimal portion of a path. More specifically, similar to how the interior points on a path do not affect the total displacement, the interior points on a surface to not affect the total surface vector.

The displacement between two points is independent of the path that connects them.

Consider for instance two paths ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ that both start at point ${\displaystyle A}$, and end at point ${\displaystyle B}$. The total displacements, ${\displaystyle \int _{r\in C_{1}}d\mathbf {r} }$ and ${\displaystyle \int _{r\in C_{2}}d\mathbf {r} }$, are both equivalent and equal to the displacement between ${\displaystyle A}$ and ${\displaystyle B}$. Note however that the total lengths ${\displaystyle \int _{r\in C_{1}}|d\mathbf {r} |}$ and ${\displaystyle \int _{r\in C_{2}}|d\mathbf {r} |}$ are not necessarily equivalent.

Similarly, given two surfaces ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ that both share the same counter-clockwise oriented boundary ${\displaystyle C}$, the total surface vectors ${\displaystyle \int _{r\in S_{1}}\mathbf {dS} }$ and ${\displaystyle \int _{r\in S_{2}}\mathbf {dS} }$ are both equivalent and are a function of the boundary ${\displaystyle C}$. This implies that a surface can be freely deformed within its boundaries without changing the total surface vector. Note however that the surface areas ${\displaystyle \int _{r\in S_{1}}|\mathbf {dS} |}$ and ${\displaystyle \int _{r\in S_{2}}|\mathbf {dS} |}$ are not necessarily equivalent.

The fact that the total surface vectors of ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ are equivalent is not immediately obvious. To prove this fact, let ${\displaystyle \mathbf {F} }$ be a constant vector field. ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ share the same boundary, so the flux/flow of ${\displaystyle \mathbf {F} }$ through ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ is equivalent. The flux through ${\displaystyle S_{1}}$ is ${\displaystyle \Phi _{1}=\int _{r\in S_{1}}\mathbf {F} \cdot \mathbf {dS} =\mathbf {F} \cdot \int _{r\in S_{1}}\mathbf {dS} }$, and similarly for ${\displaystyle S_{2}}$ is ${\displaystyle \Phi _{2}=\int _{r\in S_{2}}\mathbf {F} \cdot \mathbf {dS} =\mathbf {F} \cdot \int _{r\in S_{2}}\mathbf {dS} }$. Since ${\displaystyle \mathbf {F} \cdot \int _{r\in S_{1}}\mathbf {dS} =\mathbf {F} \cdot \int _{r\in S_{2}}\mathbf {dS} }$ for every choice of ${\displaystyle \mathbf {F} }$, it follows that ${\displaystyle \int _{r\in S_{1}}\mathbf {dS} =\int _{r\in S_{2}}\mathbf {dS} }$.

The geometric significance of the total surface vector is that each component measures the area of the projection of the surface onto the plane formed by the other two dimensions. Let ${\displaystyle \sigma }$ be a surface with surface vector ${\displaystyle \mathbf {S} =S_{x}\mathbf {i} +S_{y}\mathbf {j} +S_{z}\mathbf {k} }$. It is then the case that: ${\displaystyle S_{x}}$ is the area of the projection of ${\displaystyle \sigma }$ onto the yz-plane; ${\displaystyle S_{y}}$ is the area of the projection of ${\displaystyle \sigma }$ onto the xz-plane; and ${\displaystyle S_{z}}$ is the area of the projection of ${\displaystyle \sigma }$ onto the xy-plane.

Gauss's divergence theorem

We know that, in one dimension,

${\displaystyle \int _{a}^{b}Dfdx=f|_{a}^{b}}$

Integration is the inverse of differentiation, so integrating the differential of a function returns the original function.

This can be extended to two or more dimensions in a natural way, drawing on the analogies between single variable and multivariable calculus.

The analog of D is ∇, so we should consider cases where the integrand is a divergence.

Instead of integrating over a one-dimensional interval, we need to integrate over a n-dimensional volume.

In one dimension, the integral depends on the values at the edges of the interval, so we expect the result to be connected with values on the boundary.

This suggests a theorem of the form,

${\displaystyle \int _{V}\nabla \cdot \mathbf {u} \,dV=\int _{\partial V}\mathbf {n} \cdot \mathbf {u} dS}$

This is indeed true, for vector fields in any number of dimensions.

This is called Gauss's theorem.

There are two other, closely related, theorems for grad and curl:

• ${\displaystyle \int _{V}\nabla u\,dV=\int _{\partial V}u\mathbf {n} dS}$,

• ${\displaystyle \int _{V}\nabla \times \mathbf {u} \,dV=\int _{\partial V}\mathbf {n} \times \mathbf {u} dS}$,

with the last theorem only being valid where curl is defined.

These 3 formulas which relate volume integrals to corresponding surface integrals can be proven by decomposing a large volume into a family of infinitesimal volumes, and demonstrating each formula for an infinitesimal volume. In this case the volumes will be assumed to be infinitesimal rectangular prisms. Let ${\displaystyle R=[x_{l},x_{u}]\times [y_{l},y_{u}]\times [z_{l},z_{u}]}$ be an infinitesimal rectangular prism, where ${\displaystyle \Delta x=x_{u}-x_{l}}$, ${\displaystyle \Delta y=y_{u}-y_{l}}$, and ${\displaystyle \Delta z=z_{u}-z_{l}}$. Let ${\displaystyle (x_{c},y_{c},z_{c})\in R}$ be an arbitrary point inside the rectangular prism, and let ${\displaystyle \partial R}$ denote the outwards oriented surface of ${\displaystyle R}$.

With respect to Gauss's Divergence theorem (the relative error in the approximations vanish as ${\displaystyle \Delta _{x},\Delta _{y},\Delta _{z}\rightarrow 0^{+}}$),