Activity: Directional Derivatives

Vector Calculus I 2022
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.
What students learn Measuring slope along various directions at a point on the surface. Experimentally determining the gradient at a point on the surface. Using the Master Formula.

  1. Measurement
    1. Find the rate of change in the surface in the \(x\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{x}} = \underline{\hspace{2in}} \]
    2. Find the rate of change in the surface in the \(y\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{y}} = \underline{\hspace{2in}} \]
    3. Draw an arbitrary vector \(\boldsymbol{\vec u}\) at the blue dot on the contour mat. What are its components? \[ \boldsymbol{\vec u} = \underline{\hspace{2in}} \]
    4. Find the rate of change in the surface in the \(\boldsymbol{\vec u}\)-direction. Include units. \[ \frac{df}{ds} = \underline{\hspace{2in}} \]
  2. Computation
    1. Determine the gradient of \(f\) at the blue dot. \[ \boldsymbol{\vec{\nabla}} f = \underline{\hspace{2in}} \]
    2. Use the Master Formula to express \(\frac{df}{ds}\) in terms of \(\boldsymbol{\vec\nabla} f\), and compute the result. \[ \frac{df}{ds} =\underline{\hspace{2in}} \]
  3. Comparison
    • Compare your answers.


Copyright 2014 by The Raising Calculus Group

Instructor's Guide

Main Ideas

  • Measuring slope along various directions at a point on the surface
  • Experimentally determining the gradient at a point on the surface
  • Using the Master Formula

Prerequisite Knowledge

Students should be able to:
  • Use the measurement tool to approximate a derivative from a plastic surface

Props/Equipment

  • Tabletop Whiteboard with markers
  • Plastic surface with contour map and inclinometer
  • A handout for each student

Student Conversations

  • Some students may believe that vectors are tied to the coordinate axes, and not the contour maps; that moving the coordinate axes moves the vector along with them so that no matter the direction of the coordinate axes, the vector is always pointing in the same direction relative to those axes.
  • Some students may believe that the gradient will always point towards the top of the hill.
  • Some groups may make choose arbitrary vectors that trivialize the calculations in this activity.
  • Some students may believe that the gradient is a vector's property and not a point's property; that every vector at a given point has its own, but not necessarily unique, gradient. These students may justify their reasoning by saying that the steepest direction for any given vector must be along that vector.
  • Some students may believe that longer vectors always have a greater slope / rate of change than shorter vectors, or that vector length is what denotes magnitude of slope / rate of change; that a vector's rate of change and its magnitude represent the same value.
  • Some students may believe that every vector pointing in the same direction as the gradient is identical to the gradient vector.
  • Some students may believe that a short vector on the contour map will always correspond to a short vector on the surface.
  • Some students may believe that taking the dot product of two vectors produces a vector.
  • 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).
  • group DELETE Navigating a Hill

    group Small Group Activity

    30 min.

    DELETE Navigating a Hill
    Static Fields 2022 (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 Hill

    group Small Group Activity

    30 min.

    The Hill
    Vector Calculus II 2022 (4 years)

    Gradient

    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.
  • accessibility_new Acting Out the Gradient

    accessibility_new Kinesthetic

    10 min.

    Acting Out the Gradient
    Static Fields 2022 (6 years)

    gradient vector fields electrostatics

    Gradient Sequence

    Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
  • assignment Vector Sketch (Curvilinear Coordinates)

    assignment Homework

    Vector Sketch (Curvilinear Coordinates)
    Static Fields 2022 (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 2022 (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 Directional Derivative

    assignment Homework

    Directional Derivative

    Gradient Sequence

    Static Fields 2022 (6 years)

    You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

    1. Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
    2. In which direction in space does the water flow?
    3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
    4. Does your result to part (c) make sense from the graph?

  • assignment Contours

    assignment Homework

    Contours

    Gradient Sequence

    Static Fields 2022 (6 years)

    Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\) and \(y\) are measured in meters and that \(\mu\) is measured in kilograms. Four points are indicated on the plot.

    1. Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial y}\) at each of the four points.
    2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
    3. Evaluate the gradient of \(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\) at the point \((x,y)=(3,-2)\).

  • group Number of Paths

    group Small Group Activity

    30 min.

    Number of Paths

    E&M Conservative Fields Surfaces

    Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.
  • assignment The Gradient for a Point Charge

    assignment Homework

    The Gradient for a Point Charge

    Gradient Sequence

    Static Fields 2022 (6 years)

    The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}

    1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
    2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
    3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.


Learning Outcomes