Single, Multivariable Calculus & Classical Physics Problems

Explore the concept of functions, defining a domain and a rule that assigns a y value to each x, with y = x^2 and circle x^2 + y^2 = 1.

Learn the single-variable calculus foundations of derivatives as limits of the rate of change, using difference quotients and familiar notation like dy/dx and y'.

Visualize derivatives as the slope shown by the tiny rectangle between x and x+dx on y=f(x), where dy/dx = (y(x+dx)-y(x))/dx.

Use a key limit identity to derive the derivative of the natural logarithm. Conclude that d/dx ln x = 1/x and that e is the natural log's base.

Compute limits of f(x)/g(x) as x approaches x0 when both tend to zero, by applying derivatives f'(x0) and g'(x0) per L'Hopital's rule.

Use the derivative definition and a limit to show that the derivative of e^x is e^x, employing the natural logarithm and a z-substitution.

Apply the chain rule to differentiate composite functions by expressing df/dx as (df/dg) × (dg/dx). Explore practical examples, including e^(αx) and the derivative of αx, to illustrate the method.

Explore trigonometric identities through the complex exponential, derive cosine and sine from e^{i alpha} and e^{-i alpha}, and establish addition formulas for cos(alpha+beta) and sin(alpha+beta).

Learn to derive trigonometric derivatives by linking trigonometry with exponential functions using the chain rule, showing derivative of cosine is minus sine and derivative of sine is cosine.

Derive the derivatives of cosine and sine from the definition, using small-angle limits to justify cos' x = -sin x and sin' x = cos x.

Learn how to differentiate x^alpha for natural numbers and extend to real numbers, deriving d/dx x^alpha = alpha x^{alpha-1} via a limit and binomial expansion.

Study derivative rules for power functions: d/dx x^m = m x^(m-1) for positive integers, with a note on m=0. Extend to d/dx x^(1/m) = (1/m) x^(1/m-1) via y^m = x.

Explore how the derivative of x^alpha extends from integers to negative and rational exponents toward real numbers, using the power rule and chain rule.

Derive the product rule for two functions and extend to n functions, using the limit definition to show d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

Compute the derivative of the inverse sine on the restricted interval [-pi/2, pi/2], using dy/dx = 1/cos y and cos y = sqrt(1 - x^2), giving dy/dx = 1/sqrt(1 - x^2).

Harness Taylor series to approximate a function near a point with polynomials, using f^(n)(x0)/n! coefficients and the intuition of derivatives.

Explore how the derivative of the area function a(x) equals f(x), revealing the fundamental theorem of calculus, and see how integrating f(x) from x1 to x2 yields area.

Apply integration by parts to product f and g, showing ∫ f' g dx = f g − ∫ f g' dx and its definite form from a to b.

Explore multivariable calculus by extending theory and solving singular and multivariable exercises, with instructor Emmanuela answering your questions via YouTube solution videos.

Explore multivariable functions with multiple inputs, such as Z = f(X, Y), and visualize surfaces like circles and cones, while previewing double and triple integrals and Jacobian transformations.

Learn directional derivatives of f(x,y) by taking partial derivatives with respect to x or y, and apply the gradient dot product with a unit vector to find rate of change.

This lecture extends single-variable differentials to two variables, showing that f(x,y) changes linearly near (x0,y0) as df = f_x(x0,y0) dx + f_y(x0,y0) dy, via tangent plane approximation.

Explore the dot product's geometric meaning by relating it to the magnitudes and angle between two vectors, and see how component-wise sums yield v1 dot v2 = |v1||v2| cos alpha.

Cross product yields a vector perpendicular to the plane of two vectors, computed by determinant and the right-hand rule. Determine its magnitude as |v1||v2| sin(angle), giving parallelogram area.

Extend single-variable integration to multivariable functions by using double integrals to compute volumes over the domain of integration in the xy-plane, including constant height examples.

Compute the circle’s area via a cartesian double integral, using symmetry to take one quarter and multiply by four, yielding πR^2.

Learn how to change coordinates in double and triple integrals from cartesian to new variables using the jacobian, transforming area elements and integration boundaries.

Use polar coordinates to compute the circle's area, using the jacobian r and the relations x = r cos theta, y = r sin theta, yielding pi R squared.

Compute the volume of a sphere using a triple integral in spherical coordinates, deriving the Jacobian r^2 sin(phi) and bounds to yield 4/3 pi R^3.

Explore surface integrals as a generalization of double integrals for non-planar surfaces, using parameters u and v and the cross product to obtain the area element and unit normal.

Compute the unit normal and apply surface integral formulae for surfaces defined by F(x,y,z)=F(x,y). Project the surface onto the xy-plane and reduce to a double integral over the region.

Compute the surface area of a sphere using surface integrals in spherical coordinates, deriving the result 4 pi R^2.

Explore the divergence theorem and its proof, linking a closed surface integral of F · n dS to the triple integral of the divergence of F.

This lecture introduces line integrals along curves, using x^2 + y^2 = R^2 with x = R cos theta and y = R sin theta to compute length.

Compute the line integral along the circle x^2+y^2=R^2 by parameterizing with theta for f = (-y, x); obtain ∮ f·dl = 2π R^2 in polar coordinates, and preview Stokes theorem.

Explore Stokes' theorem, linking line integrals of a vector field to surface integrals of its curl over a boundary curve in three-dimensional space, with a hands-on proof and geometric intuition.

Apply Stokes theorem to the circle x^2+y^2=R^2 with vector field F = (-y, x, 0), equating line and surface integrals to obtain 2 pi R^2, and extend to a hemisphere with the same result.

Demonstrates applying Stokes theorem to transform integrals from the x-y plane to the u-v plane, using the Jacobian to relate area elements and the boundary line integral.

Rigorous proof of the transformation rule for double integrals when changing from (x,y) to (u,v), via the Jacobian determinant and Stokes theorem, in two dimensions.

Derive a corollary of the divergence theorem by letting F = g C with constant C, giving ∮_S G n dS = ∭ ∇G dV, a result Einstein used to estimate molecular dimensions.

apply the divergence theorem to compute the sphere's surface area, using the unit normal and divergence of the normal, and verify with the sphere's volume.

Apply the divergence theorem to compute the cube’s surface area, deriving the divergence of the normal and integrating over the cube volume, using delta functions, and obtaining 6 a^2.

Compute the surface area of a tetrahedron via a surface integral and the divergence theorem, examining each face, projection, and the role of the normal and deltas.

Learn to swap the order of integration for a double integral over the region bounded by y = 3x^2 and y = 12x, with x in [0,4].

Change the order of integration for a double integral by partitioning the region and using x = arcsin(y) and x = pi - arcsin(y), with y from 0 to 1.

Transform the region—a circle of radius 1 centered at (2,0)—using x=2+ r cos theta and y= r sin theta. Apply the jacobian r to evaluate the double integral, yielding 4/3.

Compute the volume enclosed by the paraboloid z = (x^2 + y^2)/(2a) and the sphere x^2 + y^2 + z^2 = 3a^2 using cylindrical coordinates, yielding v = (π/3)(6√3 − 5)a^3.

Compute the volume bounded by z between 0 and x + y, with xy between 1 and 2 and y between x and 2x, in the first quadrant.

Transform the Basel problem series into a double integral, then swap sum and integral. Use x as u+v and y as u−v with the jacobian to evaluate the result.

Derives the volume of an n-dimensional sphere via gaussian integrals and spherical coordinates. Yields V_n = pi^{n/2} R^n / Gamma(n/2+1) (or sqrt(pi) R^n / Gamma(n/2+1)).

Apply Stokes' theorem to convert line integral of F = (x^2 - yz, y^2 - zx, z^2 - xy) to a surface integral; curl F is zero, so line integral vanishes.

Apply Stokes' theorem to convert a line integral into a surface integral, using curl F for F = (Y, Z, X) dotted with the unit normal expressed by direction cosines.

Compute a surface integral for the outer pyramid bounded by x+y+z=a, x=0, y=0, and z=0 using parametrization and cross products. Only the plane x+y+z=a contributes, yielding a^3/2.

Use the divergence theorem to convert the surface integral over the pyramid into a volume integral of the divergence, which is 3, giving a value of a^3/2 with outward orientation.

Compute the area of a plane patch by a surface integral over the xy-projection using Z = F(X,Y), with derivatives ∂Z/∂X and ∂Z/∂Y in the integrand.

Apply the least action principle using the lagrangian L, depending on coordinates, velocities, and time. Variation of the action equals zero, yielding the lagrangian equations and Newton's law F=ma.

Derive the Hamiltonian from the Lagrangian as the energy, using time derivatives and generalized momentum; in a conservative system without explicit time dependence, energy equals kinetic plus potential energy.

Derive the Lagrangian for a gravity driven two link double pendulum by formulating the kinetic energy of both masses in terms of angles phi1 and phi2.

The lecture analyzes a simple pendulum with a horizontally moving hinge, using x and phi as Lagrangian coordinates to derive the kinetic and potential energies and the Lagrangian.

Analyze a simple pendulum attached to a rotating circle with angular frequency gamma, derive its Lagrangian, kinetic and potential energies, and the single equation of motion, with total time-derivative considerations.

Study a system of rods with symmetric configuration rotating about a vertical axis, compute kinetic and potential energies, and examine theta and omega governing motion.

Find the time-minimizing path from A to B under gravity using a Lagrangian approach, deriving the Euler-Lagrange equation and a parametric curve generated by a rolling circle.

Using the Lagrangian formalism and energy conservation, this lecture derives a Snell-like relation between theta1 and theta2 at a boundary between potentials U1 and U2, depending on mass and U2−U1.

Derive x, y, z angular momentum for a particle in cylindrical coordinates, using x = r cos theta, y = r sin theta, z, and Lz = m r^2 theta_dot.

Demonstrates helical symmetry on a z-axis helix and derives a conserved quantity combining z and theta from the lagrangian.