Lesson plan
Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If is a differentiable function and a differentiable curve in which starts at a point and ends at a point , then \int_{\gamma} \nabla\varphi(\mathbf{r})\cdot \mathrm{d}\mathbf{r} = \varphi\left(\mathbf{q}\right) - \varphi\left(\mathbf{p}\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.
Article summary
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If is a differentiable function and a differentiable curve in which starts at a point and ends at a point , then \int_{\gamma} \nabla\varphi(\mathbf{r})\cdot \mathrm{d}\mathbf{r} = \varphi\left(\mathbf{q}\right) - \varphi\left(\mathbf{p}\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.
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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.