## Vectors

• vector geometry
• assignment Flux through a Plane

assignment Homework

##### Flux through a Plane
Static Fields 2023 (4 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 $0\le x\le3$.
• assignment The Path

assignment Homework

##### The Path

Gradient Sequence

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 Vector Sketch (Curvilinear Coordinates)

assignment Homework

##### Vector Sketch (Curvilinear Coordinates)
Static Fields 2023 (2 years) 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 Vector Sketch (Rectangular Coordinates)

assignment Homework

##### Vector Sketch (Rectangular Coordinates)
vector fields Static Fields 2023 (4 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 2023 (4 years) Find the surface area of a sphere using cylindrical coordinates. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.
• group DELETE Navigating a Hill

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
Static Fields 2023 (5 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.
• group Directional Derivatives

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

Gradient Sequence

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
• assignment Cross Triangle

assignment Homework

##### Cross Triangle
Static Fields 2023 (6 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 The Hill

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 23 (4 years)

Gradient Sequence

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.
• group The Hillside

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022

Gradient Sequence

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).
• Static Fields 2023 (4 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?