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## Geometry of the Gradient

The gradient is used to relate the electric field, a vector field, to the electrostatic potential, a scalar field, by $ \vec{E} = - \vec{\nabla} V $. Many students entering into middle-division electricity and magnetism courses may have been introduced to the gradient in vector calculus and may be comfortable with calculating the gradient. However, many students may not understand the geometric meaning of the gradient which can be used to physically understand the relationship between electrostatic potential and electric field. Geometrically, the gradient is a vector which points in the direction of greatest at a particular location in a scalar field.

The dimensionality of the gradient can be a difficult concept for many students. The dimension of the gradient can be difficult because the gradient of a scalar field requires one less dimension than the scalar field itself. For example, if there is an electrostatic potential dependent on two variables, the potential can be represented as a three-dimensional surface, however, the gradient lies in the plane of the two variables. The gradient is of the same dimension as the number of variables which describe the scalar field.

For some students, the direction of the gradient can also be difficult to grasp because it is a local quantity, having a specific value at each point in space. Some students may think that the gradient points to the highest or maximum value, however, may struggle with recognizing the gradient points in direction of greatest change at a particular point in a scalar field.

### Activities

**Acting Out the Gradient***(Estimated time: 5-10 minutes)*: This kinesthetic activity introduces students to the geometric concept of the gradient through an imaginary elliptic hill in the classroom. Students use their arms to represent the gradient at their local point within the classroom. The dimension and direction of the gradient are both addressed in this activity. Many students point up to the top of the hill which can be used as an opportunity to discuss that the gradient lies in a single plane and is a local quantity which might not point to the top of the hill at a particular point.

**Navigating a Hill***(Estimated time: 30 minutes)*: 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. This activity emphasizes the gradient as a local quantity and requires students to perform calculations on a given scalar field.

**Visualizing Gradient***(Estimated time: 10-15 minutes)*: This activity utilizes a Mathematica (or Maple) worksheet to visualize the relationship between scalar fields and the gradient. Students are able to calculate and visualize gradients for various scalar fields which show the gradient is always perpendicular to the level curves of the scalar field.