Lesson plan
Surface gradient
In vector calculus, the surface gradient is a vector differential operator that is similar to the conventional gradient. The distinction is that the surface gradient takes effect along a surface. For a surface S in a scalar field u , the surface gradient is defined and notated as : \nabla_S u = \nabla u - \mathbf{\hat n} (\mathbf{\hat n} \cdot \nabla u) where \mathbf{\hat n} is a unit normal to the surface. Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface. In other words, the surface gradient is the orthographic projection of the gradient onto the surface. The surface gradient arises whenever the gradient of a quantity over a surface is important. In the study of capillary surfaces for example, the gradient of spatially varying surface tension does not make much sense; however, the surface gradient does and serves certain purposes.
Article summary
In vector calculus, the surface gradient is a vector differential operator that is similar to the conventional gradient. The distinction is that the surface gradient takes effect along a surface. For a surface S in a scalar field u , the surface gradient is defined and notated as : \nabla_S u = \nabla u - \mathbf{\hat n} (\mathbf{\hat n} \cdot \nabla u) where \mathbf{\hat n} is a unit normal to the surface. Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface. In other words, the surface gradient is the orthographic projection of the gradient onto the surface. The surface gradient arises whenever the gradient of a quantity over a surface is important. In the study of capillary surfaces for example, the gradient of spatially varying surface tension does not make much sense; however, the surface gradient does and serves certain purposes.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.