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. << Electric Field of a Line Source From Potential | Gradient Sequence |
- 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.
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