Closed curve line integrals of conservative vector fields | Multivariable Calculus | Khan Academy

TL;DR

A vector field that can be written as the gradient of a scalar field is called conservative, and the line integral between any two points in a conservative field is path independent.

Transcript

In the last video, we saw that if a vector field can be written as the gradient of a scalar field-- or another way we could say it: this would be equal to the partial of our big f with respect to x times i plus the partial of big f, our scalar field with respect to y times j; and I'm just writing it in multiple ways just so you remember what the gr... Read More

Key Insights

• 🏑 If a vector field can be expressed as the gradient of a scalar field, it is called conservative.
• 🫥 The line integral between any two points in a conservative field is path independent.
• 😚 Combining two different paths in a conservative field results in a closed line integral, which is always zero.

Summary & Key Takeaways

• A vector field that can be expressed as the gradient of a scalar field is known as conservative.

• In a conservative vector field, the line integral between any two points is independent of the path.

• Combining two different paths in a conservative field results in a closed line integral, which is always equal to zero.