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.
1. << Visualising the Gradient | Gradient Sequence | Directional Derivative >>
- Measurement
- Find the rate of change in the surface in the \(x\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{x}} = \underline{\hspace{2in}} \]
- Find the rate of change in the surface in the \(y\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{y}} = \underline{\hspace{2in}} \]
- Draw an arbitrary vector \(\boldsymbol{\vec u}\) at the blue dot on the contour mat. What are its components? \[ \boldsymbol{\vec u} = \underline{\hspace{2in}} \]
- Find the rate of change in the surface in the \(\boldsymbol{\vec u}\)-direction. Include units. \[ \frac{df}{ds} = \underline{\hspace{2in}} \]
- Computation
- Determine the gradient of \(f\) at the blue dot. \[ \boldsymbol{\vec{\nabla}} f = \underline{\hspace{2in}} \]
- Use the Master Formula to express \(\frac{df}{ds}\) in terms of \(\boldsymbol{\vec\nabla} f\), and compute the result. \[ \frac{df}{ds} =\underline{\hspace{2in}} \]
- Comparison
- Compare your answers.
Copyright 2014 by The Raising Calculus Group
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
accessibility_new Kinesthetic
10 min.
assignment Homework
assignment Homework
assignment Homework
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.
assignment Homework
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.
group Small Group Activity
30 min.
assignment Homework
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}