Lesson 403 of 1524
Vector Calculus
A vector field F is called conservative if there exists a scalar function f such that ∇ f = F
Practice this chapterA vector field F is called conservative if there exists a scalar function f such that ∇ f = F Line integrals generalize the notion of a single-variable integral to higher dimensions.
The domain of integration in a single-variable integral is a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space A vector field assigns a vector F ( x , y ) to each point ( x , y ) in a subset D of ℝ 2 or ℝ 3 .
single-variable integral — a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space. vector field — measured in ℝ 2 , an assignment of a vector F ( x , y ) to each point ( x , y ) of a subset D of ℝ 2 ; in ℝ 3 , an assignment of a vector F ( x , y , z ) to each point ( x , y …. vector field F — called conservative if there exists a scalar function f such that ∇ f = F. line integral — the integral of a function along a curve in a plane or in space.
Worked example
What does “single-variable integral” mean in Vector Calculus?
- 1Use the wording this chapter gives for single-variable integral.
- 2The book says: a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space.
- 3Do not use the meaning of vector field. That term means measured in ℝ 2 , an assignment of a vector F ( x , y ) to each point ( x , y ) of a subset D of ℝ 2 ; in ℝ 3 , an assignment of a vector F ( x , y , z ) to each point ( x , y ….
Result: a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space
Why. That is the meaning this chapter gives for single-variable integral.
Do not swap single-variable integral and vector field. single-variable integral means a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space. vector field means measured in ℝ 2 , an assignment of a vector F ( x , y ) to each point ( x , y ) of a subset D of ℝ 2 ; in ℝ 3 , an assignment of a vector F ( x , y , z ) to each point ( x , y ….
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.