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Curl or Circulation Density of a Vector Field // Vector Calculus

We've previously seen in the Vector Calculus playlist the notion of circulation or flow along a closed curve. This gave a large scale sense of how a vector field was swirling along a particular curl. But what about zooming in around any specific point? In this video we come up with a concept called circulation density or curl that measures the degree to which a miniature spinner would spin because of a vector field. That is, it tells us how much 'curl' there is at a specific point. Right now we are doing this in 2D only, but in the future we will upgrade to 3D and this will be simply the kth component of the curl. But it is useful to begin in 2D as we will segue into the next video with the immensely powerful Green's Theorem that let's us compare the circulation around the boundary to integrating the circulation density. 0:00 Visualizing Curl 3:09 Derivation 8:59 Formula MY VECTOR CALCULUS PLAYLIST: ►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line In...   OTHER COURSE PLAYLISTS: ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Log...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   ►CALCULUS I:    • Calculus I (Limits, Derivative, Integ...   ► CALCULUS II:    • Calculus II (Integration Methods, Ser...   ►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus ...   ►DIFFERENTIAL EQUATIONS:    • How to solve ODEs with infinite serie...   OTHER PLAYLISTS: ► Learning Math Series    • 5 Tips To Make Math Practice Problems...   ►Cool Math Series:    • Cool Math Series   BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based):   / treforbazett   ►Instagram (photography based):   / treforphotography