PHYS115 - Vector Calculus

The determinant of a matrix with components
$\left| \begin{matrix} î & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{matrix} \right|$
It is equal to the magnitudes of each vector multiplied by the sin of the angle between them.

The scalar projection of one vector along the other. It is equal the sum of the product of each respective component. It is also equal to the modulus of the 2 vectors multiplied by one another, and by the cosine of the angle between them.

A measure of how much a field rotates. It is the cross product of del and a vector field

A measure of how much the field spreads apart. It is the dot product of del and the vector field

A measure of the rate of change of a scalar field. It is simply del acted upon a scalar

The gradient evaluated at a point will give the greatest rate of change at that particular point

By taking the dot product of the gradient and a unit vector in the direction of interest

By taking the dot product of the gradient and an arbitrary vector, setting equal to zero, and solving. This works because a level surface is normal to the direction of maximum increase, and so the dot product must be zero.

$\int_{u_1}^{u_2}\int_{v_1}^{v_2}|\partial _u \vec{r} \times \partial _v \vec{r}|dvdu$

$\int_{u_1}^{u_2}\int_{v_1}^{v_2}\int_{w_1}^{w_2}|\partial _u \vec{r} \cdot (\partial _v \vec{r}\times \partial _w \vec{r})|dwdvdu$

Gradient is del acted upon a scalar field. It gives the direction of greatest increase of the field

Divergence is the dot product of del with a vector field. It is a measure of the quantity of flux emanating from any point in a field, and so gives information about the presence of sources or sinks. It is a quantitiave measurmenet of how much a field spreads out.

A field where the divergence is zero at all points.

Curl is the cross product of del with a vector field. It is a measure of how much the vector field twists.

A field where the curl is zero at all points

A field which may be written as the gradient of a potential

- Irrotational

Del acted upon a field twice (as given in the formula sheet)

A canonical vector $\vec{r}=x\hat{i}+y\hat{j}+f(x,y)\hat{k}$ may be found by using the constraining equation to express one coordinate in terms of the other 2.

-By finding the gradient of the level surface
- By finding $\partial_u \vec{r}\times \partial_v \vec{r}$

$x=u\cos(v), \quad y=u\sin(v), \quad z=w$

$x=r\sin(\theta)\cos(\varphi), \quad y=r\sin(\theta)\sin(\varphi), \quad z=r\cos(\theta)$

$\frac{\partial \vec{r}}{\partial u_i}$ where $u_i$ is a generalised coordinate (e.g.$x, \varphi, z$)

By taking the gradient of the inverse coordinate transformation ($u=f(x,y,z)$)

By parameterising the function along the curve and integrating with respect to the parameter

$\int_C\vec{F}.d\vec{r}$

A curve which doesnt cross itself, or wind over itself

A region R where every simple closed curve in R can be continuously shrunk to a point without leaving R. Thus, it is one with no holes

It is a generalisation of the fundamental theorem of caluclus to higher dimensions. For a simply connected region R bounded by a simple closed curve S
$\mathit{\oint_C(Pdx+Qdy)=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy}$
Where functions $\mathit{P(x,y)}$ and $\mathit{Q(x,y)}$ and their partial derivatives are finite, continuous, and single variables

By choosing functions Q and P so that $\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=1$, then the area of R may be given by $\oint_C(Pdx+Qdy)$.

By describing the surface S by a paramterised canonical position vector $\vec{r}$, the vector area element may be given by $\mathit{d\vec{S}=\partial_u\vec{r}\times\partial_v\vec{r}dudv}$. The function may thus be integrated across the bounds of both parameters.

The determinant of a matrix given by
$\mathit{\left(\begin{matrix} \frac{\partial x}{\partial u_1} & \frac{\partial y}{\partial u_1}&\frac{\partial z}{\partial u_1}\\ \frac{\partial x}{\partial u_2} & \frac{\partial y}{\partial u_2}&\frac{\partial z}{\partial u_2}\\ \frac{\partial x}{\partial u_3} & \frac{\partial y}{\partial u_3}&\frac{\partial z}{\partial u_3}\\ \end{matrix} \right)} $
It is equal to $\mathit{h_1h_2h_3}$

The length of a non-unit basis vector ($\mathit{\vec{e}_1=h_1\hat{e}_1}$)

It states that the integral of the divergence of a smooth vector field over a volume V equals the outward flux of the vector field through the boundary surface of V.
$\mathit{\int_V\nabla\cdot\vec{a}dV=\oint_S\vec{a}\cdot d\vec{s}}$

The surface integral of the curl of a function over a surface bounded by a closed path is equal to the line integral of the particular vector function around that path.
$\int_S(\nabla\times\vec{a})\cdot d\vec{S}=\oint_C\vec{a}\cdot d\vec{r}$

Imagine walking along the edge of the surface, where you are normal to the surface at that point. If the surface is to your left, then the direction in which you're walking is defined as the positive direction.

The sum of the integrals across each path gives the total curl, however it is important to consider the direction of the path.