## Cross Triangle

• assignment Flux through a Plane

assignment Homework

##### Flux through a Plane
Static Fields 2022 (3 years) Find the upward pointing flux of the vector field $\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}$ through the rectangle $R$ with one edge along the $y$ axis and the other in the $xz$-plane along the line $z=x$, with $0\le y\le2$ and $0\le x\le3$.
• group Navigating a Hill

group Small Group Activity

30 min.

##### Navigating a Hill
Static Fields 2022 (3 years)
• assignment Vector Sketch (Curvilinear Coordinates)

assignment Homework

##### Vector Sketch (Curvilinear Coordinates)
Static Fields 2022 Sketch each of the vector fields below.
1. $\boldsymbol{\vec K}=s\,\boldsymbol{\hat s}$
2. $\boldsymbol{\vec L}=\frac1s\boldsymbol{\hat\phi}$
3. $\boldsymbol{\vec M}=\sin\phi\,\boldsymbol{\hat s}$
4. $\boldsymbol{\vec N}=\sin(2\pi s)\,\boldsymbol{\hat\phi}$
• assignment Vectors

assignment Homework

##### Vectors
vector geometry Static Fields 2022 (3 years)

Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Which pairs (if any) of these vectors

1. Are perpendicular?
2. Are parallel?
3. Have an angle less than $\pi/2$ between them?
4. Have an angle of more than $\pi/2$ between them?

• assignment Vector Sketch (Rectangular Coordinates)

assignment Homework

##### Vector Sketch (Rectangular Coordinates)
vector fields Static Fields 2022 (3 years) Sketch each of the vector fields below.
1. $\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}$
2. $\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}$
3. $\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}$
• assignment Sphere in Cylindrical Coordinates

assignment Homework

##### Sphere in Cylindrical Coordinates
Static Fields 2022 (3 years) Find the surface area of a sphere using cylindrical coordinates.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)
Quantum Fundamentals 2022

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
• assignment Divergence through a Prism

assignment Homework

##### Divergence through a Prism
Static Fields 2022 (4 years)

Consider the vector field $\vec F=(x+2)\hat{x} +(z+2)\hat{z}$.

1. Calculate the divergence of $\vec F$.
2. In which direction does the vector field $\vec F$ point on the plane $z=x$? What is the value of $\vec F\cdot \hat n$ on this plane where $\hat n$ is the unit normal to the plane?
3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

• assignment Polar vs. Spherical Coordinates

assignment Homework

##### Polar vs. Spherical Coordinates
Central Forces 2022 (2 years)

Show that the plane polar coordinates we have chosen are equivalent to spherical coordinates if we make the choices:

1. The direction of $z$ in spherical coordinates is the same as the direction of $\vec L$.
2. The $\theta$ of spherical coordinates is chosen to be $\pi/2$, so that the orbit is in the equatorial plane of spherical coordinates.

• assignment Volume Charge Density

assignment Homework

##### Volume Charge Density
Static Fields 2022 (4 years)

Sketch the volume charge density: $$\rho (x,y,z)=c\,\delta (x-3)$$

• Static Fields 2022 (4 years)

Use the cross product to find the components of the unit vector $\mathbf{\boldsymbol{\hat n}}$ perpendicular to the plane shown in the figure below, i.e.  the plane joining the points $\{(1,0,0),(0,1,0),(0,0,1)\}$.