Flux through a Plane

    • 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.
    • 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 Vectors

      assignment Homework

      Vectors
      vector geometry 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?

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

    • group Directional Derivatives

      group Small Group Activity

      30 min.

      Directional Derivatives
      Vector Calculus I 2022

      Directional derivatives

      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.
  • 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 \(0\le x\le3\).