Multivariable Calculus (Calc III) - Complete Semester Course - Conservation of Energy, Multivariable Calculus

We look at the conservation of total energy in Newtonian mechanics. Namely, suppose an object of mass m travels according to r(t), a≤t≤b with a background vector field F. We discuss kinetic and potential energy in relation to the work done by F. Here are some details: - We define kinetic energy as 1/2 m ||r'(t)||^2. - We discuss the work performed by a background vector field F on the moving mass, using a vector line integral for calculation, emphasizing how the change in kinetic energy relates to the work done by the vector field. - Looking at potential energy, we focus on its dependence on the object's position in relation to the vector field. - We explain conservative vector fields as those where the vector field equals the negative gradient of the potential energy function. - Finally, we apply the fundamental theorem for line integrals to demonstrate how the conservation of total energy is upheld, showing the equality between the sum of kinetic and potential energy at the beginning and end of the motion. This video is Multivariable Calculus Unit 6 Lecture 11. #calculus #multivariablecalculus #mathematics #iitjammathematics #physics #NewtonianMechanics #VectorFields #LineIntegrals #calculus3