Calculus IV: Vector Calculus (Line Integrals, Surface Integrals, Vector Fields, Greens' Thm, Divergence Thm, Stokes Thm, etc) **Full Course** The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus
The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus Transcript and Lesson Notes
Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what hap
Quick Summary
Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what hap
Key Takeaways
- Review the core idea: Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what hap
- Understand how Solution fits into The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus.
- Understand how Example fits into The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus.
- Understand how math fits into The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus.
- Understand how fundamental theorem of line integrals fits into The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus.
Key Concepts
Full Transcript
Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what happened in the middle did not matter. In this video we upgrade to the Fundamental Theorem of Line Integrals, which is a generalization of the Fundamental Theorem of Calculus. It says that when you take the line integral of a conservative vector field (ie one where the field can be written as the gradient of a scalar potential function), then this line integral is similarly just the difference of the function at the endpoints and is thus path independent - only the endpoints matter. In this video we will motivate this theorem, prove it formally, and connect the idea back to that of conservative vector fields. 0:00 FToC 1:24 Fundamental Thm of Line Integrals 2:50 Proof 5:33 Conservative Fields MY VECTOR CALCULUS PLAYLIST: ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa OTHER COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography
Lesson FAQs
What is The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus about?
Back in 1st year calculus we have seen the Fundamental Theorem of Calculus II, which loosely said that integrating the derivative of a function just gave the difference of the function at the endpoints. That is, what hap
What key concepts are covered in this lesson?
The lesson covers Solution, Example, math, fundamental theorem of line integrals, FToC.
What should I learn before The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus?
Review the previous lessons in Calculus IV: Vector Calculus (Line Integrals, Surface Integrals, Vector Fields, Greens' Thm, Divergence Thm, Stokes Thm, etc) **Full Course**, then use the transcript and key concepts on this page to fill any gaps.
How can I practice after this lesson?
Practice by applying the main concepts: Solution, Example, math, fundamental theorem of line integrals.
Does this lesson include a transcript?
Yes. The full transcript is visible on this page in indexable HTML sections.
Is this lesson free?
Yes. CourseHive lessons and courses are available to learn online for free.
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