Y2T1 Calculus

What is the Einstein summation convention?

Any repeated suffix in a product is implicitly summed over its range. In 3D, aᵢxᵢ means sum over i = 1 to 3 aᵢxᵢ.

What is the Kronecker delta δᵢⱼ?

δᵢⱼ = 1 if i = j, and δᵢⱼ = 0 if i ≠ j.

What is the contraction identity for the Kronecker delta?

δᵢⱼ aⱼ = a_i. A repeated index is summed, leaving the free index i.

What is a contraction?

An operation in which repeated indices are summed, reducing the number of free indices; e.g. δᵢⱼ aⱼ = a_i.

What is the permutation symbol εᵢⱼₖ?

εᵢⱼₖ = 0 if any two indices are equal; +1 if (i,j,k) is a cyclic permutation of (1,2,3); −1 if it is an acyclic/anti-cyclic permutation.

What are ε_123, ε_321 and ε_133?

ε_123 = 1, ε_321 = −1, and ε_133 = 0.

State the ε–δ identity in one common form.

εᵢⱼₖ ε_klm = δᵢₗ δⱼₘ − δᵢₘ δⱼₗ.

State the alternative ε–δ identity obtained by relabelling indices.

εᵢⱼₖ ε_ilm = δⱼₗ δₖₘ − δⱼₘ δₖₗ.

What is the vector product a × b?

The vector perpendicular to both a and b, with direction given by the right-hand rule. It is represented by the determinant |i j k; a1 a2 a3; b1 b2 b3|.

When is a × b = 0?

Exactly when a and b are parallel (including when one is the zero vector).

What is the antisymmetry property of the cross product?

a × b = −(b × a).

Give the index notation for the cross product.

[a × b]_i = εᵢⱼₖ aⱼ bₖ.

What is the scalar product?

a · b = aᵢbᵢ = a_1b_1 + a_2b_2 + a_3b_3.

When are two vectors orthogonal?

They are orthogonal when a · b = 0.

What is the triple scalar product in index notation? a . (b x c)

a · (b × c) = εᵢⱼₖ a_i b_j c_k.

What does a · (b × c) represent geometrically?

The signed volume of the parallelepiped formed by a, b and c. It is zero iff a, b and c are coplanar.

State the cyclic interchange identity for the triple scalar product.

a · (b × c) = (a × b) · c.

State the triple vector product identity.

a × (b × c) = (a · c)b − (a · b)c.

What is a level surface of a scalar field φ?

A surface on which φ is constant: φ(x,y,z) = constant.

What is the directional derivative ∂φ/∂s?

The rate of change of φ in the direction of the unit vector ŝ: ∂φ/∂s = ŝ · ∇φ.

In which direction is the directional derivative maximised?

It is maximised in the direction of ∇φ, normal to the level surface φ = constant.

What is the gradient of φ?

The vector giving the direction of maximum increase of φ and whose magnitude is the maximum directional derivative: ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k.

What is the Cartesian formula for the gradient?

∇φ = i ∂φ/∂x + j ∂φ/∂y + k ∂φ/∂z.

What is the cylindrical-polar formula for the gradient?

∇φ = r̂ ∂φ/∂r + θ̂(1/r)∂φ/∂θ + k ∂φ/∂z.

What is the geometric relationship between ∇φ and φ = constant?

∇φ is normal to the level surface φ = constant.

What is the tangent-plane equation to φ = φ(P) at P?

(r − r_P) · (∇φ)_P = 0.

Write the tangent-plane equation in Cartesian components.

(∂φ/∂x)_P(x−x_P) + (∂φ/∂y)_P(y−y_P) + (∂φ/∂z)_P(z−z_P) = 0.

What is the divergence of a vector field A?

div A = ∇ · A.

What is the curl of a vector field A?

curl A = ∇ × A.

What is the Cartesian formula for divergence?

∇ · A = ∂A₁/∂x + ∂A₂/∂y + ∂A₃/∂z.

What is the Cartesian formula for curl?

∇ × A = (∂A₃/∂y − ∂A₂/∂z)i + (∂A₁/∂z − ∂A₃/∂x)j + (∂A₂/∂x − ∂A₁/∂y)k.

What linearity identities does ∇ satisfy?

∇(φ+ψ)=∇φ+∇ψ; div(A+B)=div A+div B; curl(A+B)=curl A+curl B.

State the product rule for a gradient.

∇(φψ) = φ∇ψ + ψ∇φ.

State the product rule for divergence.

div(φA) = φ div A + ∇φ · A.

State the product rule for curl of a scalar times a vector.

curl(φA) = φ curl A + ∇φ × A.

State the divergence of a cross product identity.

div(A × B) = B · curl A − A · curl B.

State the curl of a cross product identity.

curl(A × B) = (B · ∇)A − B div A − (A · ∇)B + A div B.

State the gradient of a dot product identity.

∇(A · B) = (B · ∇)A + (A · ∇)B + B × curl A + A × curl B.

What is the Laplacian of a scalar field?

∇²φ = div(∇φ) = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² in Cartesian coordinates.

What is the Laplacian of a vector field in Cartesian coordinates?

∇²A = ∂²A/∂x² + ∂²A/∂y² + ∂²A/∂z², applied componentwise.

What is Laplace’s equation?

∇²φ = 0.

What is the curl of a gradient?

curl(∇φ) = 0, provided φ has continuous second derivatives.

What is the divergence of a curl?

div(curl A) = 0, provided A has continuous second derivatives.

State the curl-of-curl identity.

curl(curl A) = ∇(div A) − ∇²A.

What is a scalar field?

A scalar function defined at every point of a region of space.

What is a vector field?

A vector function defined at every point of a region of space.

What is an irrotational vector field?

A vector field A for which curl A = 0.

What is a solenoidal vector field?

A vector field A for which div A = 0.

What is the position vector in Cartesian coordinates?

r = xi + yj + zk.

What are div r and curl r?

div r = 3 and curl r = 0.

What is the magnitude of the position vector?

r = |r| = (x²+y²+z²)^(1/2).

What is ∇r for r = |r|?

∇r = r̂ = r/r, for r ≠ 0.

What is the path integral of f along a curve γ?

The limit of sums Σ f_n δs_n as the partition is refined: ∫_γ f ds = lim Σ f_n δs_n.

Can f in ∫_γ f ds be scalar or vector-valued?

Yes. A path integral may integrate either a scalar or a vector function.

What is the tangent vector to a curve?

t̂ = dr/ds = lim_{δs→0}(δr/δs), a unit vector tangent to the curve.

What is the path element?

dr = t̂ ds.

How is a vector line integral defined?

∫γ F · dr = ∫γ F · t̂ ds.

What is a conservative vector field?

A field F for which ∮_γ F · dr = 0 for every closed curve γ in the region.

What is a potential of a vector field F?

A scalar field φ such that F = ∇φ.

If F = ∇φ, how is ∫_A^B F · dr evaluated?

∫_A^B F · dr = φ(B) − φ(A), independent of the path.

What is the fundamental theorem for line integrals?

If F = ∇φ, then ∫_γ F · dr = φ(B) − φ(A). Consequently, every closed-loop integral is zero.

What is the relationship between conservativeness and path independence?

For suitable fields, F is conservative iff line integrals are path-independent; equivalently, all closed-loop integrals vanish.

What is the practical parametrisation formula for a vector line integral?

If r=r(t), t∈[t₀,t₁], then dr/dt dt and ∫γ F·dr = ∫{t₀}^{t₁} F(r(t))·r'(t) dt.

How is a piecewise path integral evaluated?

Split the path into segments and add the integrals over each segment.

What is a surface integral?

The limit of sums Σ f_n δS_n as the surface partition is refined: ∫_S f dS = lim Σ f_n δS_n.

Can the integrand in a surface integral be scalar or vector-valued?

Yes.

What is a closed surface?

A surface that divides 3D space into two non-connected regions: an interior and an exterior.

What is a convex surface?

A surface crossed by any straight line at most twice.

What is an open surface?

A surface that does not divide space into interior and exterior regions; it has a rim represented by a closed curve.

What is the vector area element?

dS_vector = n̂ dS, where n̂ is the unit normal.

For a plane region in the xy-plane, what is dS?

dS = dx dy.

How is a plane surface region represented when bounded by y=F₁(x) and y=F₂(x)?

S = {(x,y): a≤x≤b, F₂(x)≤y≤F₁(x)}.

How is a scalar surface integral over that region evaluated?

∫S f dS = ∫_a^b ∫{F₂(x)}^{F₁(x)} f(x,y) dy dx.

How can the order of integration be reversed for a plane region?

If x=G₁(y) is the right boundary and x=G₂(y) the left boundary, then integrate ∫c^d∫{G₂(y)}^{G₁(y)} f(x,y) dx dy.

How are a plane area S and its projection Σ onto the xy-plane related?

Σ = S cosθ, where cosθ = |n̂·k|.

What is the differential projection formula onto the xy-plane?

dΣ = |n̂·k| dS, so dS = dx dy/|n̂·k|.

State the projection theorem onto z=0.

∫S f(P)dS = ∫Σ f(P) dxdy/|n̂·k|, where Σ is the projection of S onto z=0.

If S is z=φ(x,y), how does the projection theorem simplify?

∫S f(x,y,z)dS = ∫Σ f(x,y,φ(x,y))√(1+φ_x²+φ_y²) dxdy.

What is a volume integral?

The limit of sums Σ f_n δτn as the volume partition is refined: ∫τ f dτ = lim Σ f_n δτ_n.

Can the integrand in a volume integral be scalar or vector-valued?

Yes.

What is the Cartesian volume element?

dτ = dx dy dz.

State Green’s theorem in the plane.

For a positively oriented, simple closed boundary C of R, ∮_C(L dx+M dy)=∫_R(∂M/∂x−∂L/∂y)dxdy.

What orientation is used in Green’s theorem?

The boundary C is traversed counter-clockwise, so the region remains on the left.

What is the 2D Stokes theorem form of Green’s theorem?

For F=Li+Mj, ∮_C F·dr = ∫_R curl F·dS, where curl F=(∂M/∂x−∂L/∂y)k.

What is the 2D divergence theorem form of Green’s theorem?

∫_R div F dxdy = ∮_C F·n̂ ds, with the outward normal to C.

For F=M i−L j, what is div F?

div F = ∂M/∂x − ∂L/∂y.

What is the outward normal line element in 2D?

n̂ ds = dy i − dx j for a positively oriented boundary.

What is the area enclosed by a simple closed plane curve?

Area(R) = (1/2)∮_C(x dy − y dx), with C counter-clockwise.

How does Green’s theorem extend to non-convex regions?

Divide the region into simple convex subregions. Apply Green’s theorem to each; integrals over internal dividing curves cancel.

What does simply-connected mean for a planar region?

Every closed curve in the region can be continuously shrunk to a point without leaving the region.

What does multiply-connected mean?

The region is not simply-connected; in 2D this typically means it contains one or more holes.

How is Green’s theorem oriented on a multiply-connected region?

Use the entire boundary, with the region always on the left: outer boundaries are counter-clockwise and inner boundaries are clockwise.

State Green’s theorem for a multiply-connected region.

∮_C(L dx+M dy)=∫_R(∂M/∂x−∂L/∂y)dxdy, where C includes every boundary component with the region on the left.

grad of a vector doesn’t make sense in this course!! why?

grad applies to scalars in this course. grad of a vector would create a tensor product