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Multivariable calculus, class #29: Scalar line integrals
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Mathematician spotlight: Steve Robinson
We play a small game of Hex, note that there is always a winner, and that this has surprising applications to higher mathematics. We review the meaning of curl and divergence of a vector field, and discuss which related expressions make sense and which do not. We show that the curl of a gradient vector field is always 0, and that the divergence of the curl of any vector field is 0. We give an expression for computing a scalar line integral, and discuss what it means. We do an example of a scalar line integral over part of a paraboloid, and one over a helix. We discuss that when you integrate the function "1" you get the length of the curve, which is just our arc length formula from earlier.
We play a small game of Hex, note that there is always a winner, and that this has surprising applications to higher mathematics. We review the meaning of curl and divergence of a vector field, and discuss which related expressions make sense and which do not. We show that the curl of a gradient vector field is always 0, and that the divergence of the curl of any vector field is 0. We give an expression for computing a scalar line integral, and discuss what it means. We do an example of a scalar line integral over part of a paraboloid, and one over a helix. We discuss that when you integrate the function "1" you get the length of the curve, which is just our arc length formula from earlier.
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