## Flux through a Plane

• 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.
• 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 The Path

assignment Homework

##### The Path

Vector Calculus I 2022 You are climbing a hill along the steepest path, whose slope at your current location is $1\over5$. There is another path branching off at an angle of $30^\circ$ ($\pi\over6$). How steep is it?
• assignment Cross Triangle

assignment Homework

##### Cross Triangle
Static Fields 2022 (5 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 $\{(1,0,0),(0,1,0),(0,0,1)\}$.

• group DELETE Navigating a Hill

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
Static Fields 2022 (4 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
• assignment Curl Practice including Curvilinear Coordinates

assignment Homework

##### Curl Practice including Curvilinear Coordinates

Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

1. $$\vec{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\vec{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\vec{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\vec{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\vec{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\vec{K} = s^2\,\hat{s}$$
7. $$\vec{L} = r^3\,\hat{\phi}$$

• group The Hillside

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
• group Directional Derivatives

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

• 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 $0\le x\le3$.