Beauty of Line Integral (Calculus) .

you would encounter scholar line integral and vector line integrals in your calculus book let's understand what they mean here i would use a rectangle that has an infinite decimal with and possibly some height to get intuitive meaning of scalar line integral consider an equation of a curve the integral of this curve with respect to x axis is given by this formula it got variable height an infinitesimal width of a small rectangle let's call it a stick right now we only got four stick let's keep on adding them if you do so you approximate the area under this curve just by using 63 sticks you can easily approximate the area the area would be more precise if our stick possesses infinitesimal width and we increase the number of sticks this is also one kind of line integral here we integrate the curve with respect to x axis now let's try to add those sticks on a 2d curve and try to integrate the function here is a parametrized equation of a circle in x y plane now let's plot another curve orthogonal to this circle here every point of this circle is projected upward which builds this beautiful function the line integral of this function along the curve gives the area of this blue surface just imagine adding those tiny stick along the path of a circle now here is an interesting thing if you change this formula's height function to some constant value let's make it one then you will get the area of this cylinder in this case every stick got the height one its height is directly proportional to the magnitude of this constant okay here is another curve parameterized in x y plane again let's draw another curve orthogonal to this parameterized curve see how this curve is parameterized the line integral of this function where we add tiny sticks along the parameterized curve in x y plane gives the area of this surface please be careful with the negative area from the illustration we see that the graph dips below the x y plane there is technique to calculate area back in single variable calculus to handle this type of negative and positive function now here is an another example a parameterized ellipse in xy plane this time let's draw a surface that surrounds this ellipse as you can see this surface surrounds our ellipse in all way let's try to solve this line integral problem first we need to project our ellipse over the surface now the height of stick is given by this parameterized function now we don't want the surface let's remove it evaluating this line integral is like adding stick along the path of ellipse that has height generated by this parameterized function you can convert this problem into this did you notice how important parameterizing a curve is it just makes integral more easier to compute here notice the negative area be sure to take it into account while calculating the area under this graph line integral of scalar function along a curve in space is just not clear what such an integral mean but they are good for more than just computing areas consider this spiral curve as a spring let's say the density of this spring varies in space according to this scanner function just notice that the density of this spring increases as the spring spirals up the z-axis if this density function is mass per unit length then the mass can be calculated by using this formula that's some geometrical meaning of a line integral of scalar functions now let's compute line integral of a vector field i have plotted a force field given by this vector function now let's draw a curve parameterized in this force field these force field also acts on every points on this curve also a curve in space is oriented according to their unit tangent vectors and normal vectors for computing line integral of a vector field along the path of this curve we only need vector field and unit tangent vectors that acts on this curve projecting this vector field in the direction of unit tangent vectors of our curve is a dot product by doing so we can get tangential component of forces on every point for evaluating line integral we need to add every value of this tangential forces along the length of curve so we multiply every of these tangential vectors with the length of curve when a curve starts and ends at the same point it is a closed curve or loop now let's suppose that this field represents a velocity field under these circumstances the integral of velocity vectors and unit tangent vectors along a curve in the region gives the circulation around the curve this is for 2d curve where is the rate at which a fluid is leaving or entering a region enclosed by a closed curve gives the flux this is the integral of velocity vectors and unit normal vectors along a curve notice the difference between flux and circulation flux is integral of normal component of vector field circulation is integral of tangential component of vector field this vector field can be anything velocity field force field magnetic field electric field the logic is same okay that's some definition of vector line integral now let's find the work done by this force field on a particle as it moves along the helix given by this parameterized equation from the initial point to the final we can evaluate the work done by using this line integral first let's get the force field acting only on curve points now you can remove other unwanted force field now compute this dot product project every force field in the direction of unit tangent vectors and multiply its magnitude with the arc length of curve it's difficult to compute using this integral formula but we can convert it into more simpler form there exists some relation between direction vectors of curve unit tangent vectors and arc length finally compute the line integral and congrats you got the work done you

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Length: 8min 56sec (536 seconds)

Published: Mon Jul 11 2022