Math for Electricity and Magnetism

Explain the scalar (dot) product of two vectors, showing projection onto a unit vector i-hat as dot product magnitude times cos theta, with zero projection for perpendicular vectors.

Explore the dot product of two vectors, defined as |a||b|cos theta, and learn its properties, including commutativity, distributivity, and behavior for perpendicular or parallel vectors.

Use the component form of the scalar dot product with the standard basis i, j, k to compute A·B = a_x b_x + a_y b_y + a_z b_z.

This lecture presents orthonormality relations in the standard basis e1, e2, e3, showing e_i dot e_j equals one when i=j and zero otherwise, and introduces a related notation.

Explore the scalar product in component representation, showing a·b = a1 b1 + a2 b2 + a3 b3 and using column notation and the Kronecker delta.

Learn to compute the flux of a vector through a surface using the surface normal and dot product; for a constant vector it reduces to e dot s.

Compute the cross product a cross b, with magnitude |a||b|sin(theta) and a unit perpendicular vector. Apply the right-hand rule to determine its direction.

The magnitude of the vector product equals the area of the parallelogram spanned by vectors a and b, |a| |b| sin φ, with a unit vector perpendicular to the plane.

Explore the properties of the vector product, including the magnitude |a×b| = |a||b|sin(theta), its zero values at parallel or anti-parallel vectors, and its maximum at 90 degrees.

Explore cross product rules: non-commutativity (a × b = − b × a), distributivity, and no associativity; apply the right-hand rule to get unit vector and magnitude |A||B| sin φ.

Compute the cross product of two vectors using a determinant, with the standard basis and Cartesian components to form the determinant and obtain the i, j, k terms.

Learn to compute the triple scalar product in component form using Cartesian x, y, z, derive the cross product via a determinant representation, and relate it to dot products.

Understand that the triple scalar product gives the volume formed by the three vectors, computed as |a·(b×c)|, with the orientation set by the right-hand rule.

Study the triple scalar product a · (b × c); it vanishes for parallel vectors or when a is perpendicular to b × c, and swapping rows flips the sign.

Express a point’s position in Cartesian coordinates with x, y, z and vectors i, j, k in an orthonormal basis, r = x i + y j + z k.

Explore cartesian coordinates with the unit vectors i, j, k in a right-handed, orthogonal, position-independent basis. Learn why this constancy aids symmetry-based problem solving and glimpse cylindrical and spherical coordinates.

Describe how surfaces x = constant, y = constant, and z = constant intersect to define a point in Cartesian coordinates, with projections on the xy plane illustrating constant coordinates.

Explore how Cartesian coordinates connect to spherical coordinates to locate a point, using radius, z, and azimuth phi; observe how fixing z and phi traces circles and lines.

Explore the cylindrical basis in electromagnetism: define the local unit vectors e_rho, e_phi, and e_z, relate them to Cartesian units, and visualize surfaces of constant rho, phi, and z.

Note three mutually orthogonal unit vectors define the position vector r, with e_r and e_phi depending on phi, and Cartesian coordinates have constant unit vectors.

Relate cartesian coordinates to cylindrical coordinates using x = r cos phi, y = r sin phi, with r = sqrt(x^2 + y^2), phi = arctan(y/x), and z remains unchanged.

Express cylindrical basis in the Cartesian frame. Derive e_rho = cos(phi) e_x + sin(phi) e_y and e_phi = -sin(phi) e_x + cos(phi) e_y with phi = arctan(y/x) and z unchanged.

Express the position vector in Cartesian coordinates x, y, z and relate its components to cos(phi) and sin(phi) projections, while illustrating partial derivative notation and unit vector magnitude.

Express e_phi in the cartesian basis through the second method by differentiating r with respect to phi, yielding e_phi = -sin(phi) i + cos(phi) j with magnitude 1.

We express e_z in the cartesian basis using the second method; with phi constant, the partial derivative with respect to z is zero, yielding e_z as a unit vector.

Show how complex notation e^{i phi} encodes the cosine and sine of an angle, with e^{i phi} = cos(phi) + i sin(phi). Illustrate how half-angle forms relate to these components.

remark 3 examines cosine and sine terms involving phi, derives relationships via the partial derivative with respect to phi, and notes how using ephi simplifies the expressions.

Derive the rotation matrix for coordinates, showing the 3×3 form with cosφ and sinφ, and verify row-by-column products link the rotated basis to g and k.

Explore the rotation matrix in part 2, including projecting vectors onto axes, using cosines and sines, and review of matrix transpose and its properties.

Explore rotation matrices in three dimensions, showing that the transpose equals the inverse for 3×3 matrices and how identity relationships arise from a three-by-three basis.

Examine spherical coordinates, linking the radius r and angles theta and phi to a point in three-dimensional space, and clarify the range and interpretation of r, theta, and phi.

Explore how holding rho constant yields a sphere, how varying phi traces circles and a cone, and how phi constant yields a plane.

Explore the spherical basis in three-dimensional space, defining the position vector r with unit vectors e_r, e_theta, and e_phi, and explain their normal and tangential relationships and right-handed orientation.

Examine how three mutually orthogonal unit vectors e_r, e_theta, and e_phi form a local spherical coordinate basis, with the position vector r e_r as a function of theta and phi.

Show how to express cylindrical basis vectors er and e_phi in a plane and relate them to r and theta, and derive partial derivatives with respect to r and theta.

demonstrates projecting a vector in a spherical basis using dot products and cross products, deriving components along er and e_phi, and computing partial derivatives with respect to phi and theta.

Recognize that the Cartesian coordinate system has constant unit vectors, whereas spherical coordinates do not; derive the relations between Cartesian and spherical coordinates by resolving unit vectors into Cartesian components.

This lecture explains how cartesian coordinates relate to spherical coordinates using a right triangle with r, theta, and phi, deriving z from r cos(phi) and x, y from r sin(phi).

Convert cartesian coordinates to spherical coordinates by deriving r = sqrt(x^2 + y^2 + z^2), z = r cos(phi), and phi = arctan(y/x).

Explain expressing the position vector in Cartesian form x i + y j + z k and in spherical form as r e_r with unit vectors e_r, e_theta, e_phi.

Explore how unit vectors in spherical coordinates relate to each other, expressing e_r, e_theta, and e_phi as functions of theta and phi with their directional cosines.

Derive how the spherical unit vectors r̂, θ̂, and φ̂ express in the Cartesian basis Ex, Ey, and k̂, using cosφ and sinφ relations.

Explore the relationship between Cartesian and spherical coordinates and analyze partial derivatives with respect to theta and phi, including cosine and sine components.

Explore expressing the position vector in spherical coordinates and differentiating with respect to r, theta, and phi, revealing unit vectors and magnitudes.

Remark 3 explains how cosigner states combine with pi over two shifts to yield sinus state or cosigner state, showing sign relations that map between these states.

Derive the rotation matrix for spherical coordinates by relating the spherical basis to cartesian components, using cosines and sines of theta and phi, and show the transpose equals the inverse.

Learn how to compute differential of a multivariable function using df = f_x dx + f_y dy + f_z dz for independent x, y, z, with thermodynamics and electromagnetism examples.

Define the infinitesimal displacement vector in Cartesian coordinates by combining dx, dy, dz into ds, and relate it to infinitesimal line and area elements.

Explore the infinitesimal volume element in Cartesian coordinates, expressed as dx dy dz to form a small cube and dv with cubic meters.

Explore the infinitesimal displacement vector dl in cylindrical coordinates, using e_r and e_φ, and derive the polar area element dA = r dr dφ with arc length r dφ.

Explore the area element for the cylinder’s lateral surface using polar coordinates, deriving the small arc and differential element ds with radius r, phi, and z.

Explore the infinitesimal volume element in cylindrical coordinates, using rho, phi, and z to form dv, and recognize its cubic meter units across three differential components.

Explain the infinitesimal displacement vector in spherical coordinates, deriving dl = e_r dr + r de_r and showing how theta and phi derivatives of r e_r determine the displacement.

Examine the area element on a sphere using spherical coordinates, combining infinitesimal arcs to form a small area; confirm the two-dimensional surface has unit meter squared.

Learn how to express the infinitesimal volume element in spherical coordinates, derive dv from the three differentials, and relate it to Cartesian and cylindrical forms.

Explore the gradient operator and how it describes the spatial variations of a scalar field via partial derivatives, in Cartesian and spherical coordinates, alongside related vector operators.

the lecture demonstrates the gradient operator in cartesian coordinates, defines the position vector and its magnitude, and derives ∇r = (x/r, y/r, z/r) for r = sqrt(x^2+y^2+z^2).

Compute gradient operator components in Cartesian coordinates by differentiating the position vector r with respect to x, y, and z, yielding x/r, y/r, z/r, and identify the unit vector.

Explore the gradient operator's linearity, apply it to functions in cartesian coordinates, and derive the gradient of r = sqrt(x^2 + y^2 + z^2) and 1/r.

Compute the gradient of 1/r in Cartesian coordinates, show ∇(1/r) = -r / r^3, and connect the result to the position vector and electrostatics applications.

Explore how the divergence operator acts on a vector field to yield a scalar, using Cartesian form ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z and spherical components A_r, A_θ, A_φ.

Navigate the nabla operator, and apply gradient, divergence, and curl to vector fields in Cartesian and spherical coordinates, with component forms like A_x, A_y, and A_z.