Mr. Integral
Math notes/Calculus III/Vector Calculus

THE WHOLE UNIT · ONE REFERENCE

Vector Calculus
Cheat sheet.

The key rules, formulas and reminders from all 9 topics, gathered into reference cards.

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Key formulas, conditions and traps · Read down each column.

Scalar and Vector Line Integrals

Core rule

ds=∥r′∥dt,ds=\|\mathbf r'\|dt, dr=r′dt.d\mathbf r=\mathbf r'dt.

Watch for

Scalar line integrals use speed. Work uses a dot product and an oriented path.

Conservative Fields and Potentials

Core rule

F=∇ϕ  ⟹  W=ϕ(B)−ϕ(A).\mathbf F=\nabla\phi\implies W=\phi(B)-\phi(A).

Watch for

Curl zero needs a suitable domain for the converse. Check holes and singularities.

Green’s Theorem

Core rule

∮P dx+Q dy=∬(Qx−Py) dA.\oint P\,dx+Q\,dy=\iint(Q_x-P_y)\,dA.

Watch for

Keep the region on the left and check smoothness throughout its interior.

Divergence and Curl

Core rule

div⁡F=Px+Qy+Rz.\operatorname{div}\mathbf F=P_x+Q_y+R_z. curl⁡F=(Ry−Qz,Pz−Rx,Qx−Py).\operatorname{curl}\mathbf F=(R_y-Q_z,P_z-R_x,Q_x-P_y).

Watch for

Divergence is scalar expansion; curl is vector rotation. Neither equals speed.

Surface Area and Scalar Surface Integrals

Core rule

dS=∥ru×rv∥du dv.dS=\|\mathbf r_u\times\mathbf r_v\|du\,dv.

Watch for

Include the area scale, distinguish a wall from a closed surface, and avoid covering twice.

Oriented Surface Flux

Core rule

∬SF⋅n dS=∬DF(r)⋅(ru×rv)du dv.\iint_S\mathbf F\cdot\mathbf n\,dS=\iint_D\mathbf F(\mathbf r)\cdot(\mathbf r_u\times\mathbf r_v)du\,dv.

Watch for

Check orientation before integrating. The cross product already includes the area scale.

The Divergence Theorem

Core rule

∬∂VF⋅n dS=∭V∇⋅F dV.\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS=\iiint_V\nabla\cdot\mathbf F\,dV.

Watch for

Require a closed outward boundary and a smooth field throughout the enclosed solid.

Stokes’ Theorem

Core rule

∮∂SF⋅dr=∬S(∇×F)⋅n dS.\oint_{\partial S}\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS.

Watch for

Use the right-hand rule. Integrate curl, not the original field, across the surface.

Calculus III: Engineering Checkpoint

Core rule

Match the quantity to its domain: rate, curve work, surface flux or volume accumulation.

Watch for

Check orientation, Jacobians, units, smoothness and boundary conditions before choosing a theorem.

01

Scalar and Vector Line Integrals

2 reference blocks

Read lesson ↗

Core rule

ds=∥r′∥dt,ds=\|\mathbf r'\|dt, dr=r′dt.d\mathbf r=\mathbf r'dt.

Watch for

Scalar line integrals use speed. Work uses a dot product and an oriented path.

02

Conservative Fields and Potentials

2 reference blocks

Read lesson ↗

Core rule

F=∇ϕ  ⟹  W=ϕ(B)−ϕ(A).\mathbf F=\nabla\phi\implies W=\phi(B)-\phi(A).

Watch for

Curl zero needs a suitable domain for the converse. Check holes and singularities.

03

Green’s Theorem

2 reference blocks

Read lesson ↗

Core rule

∮P dx+Q dy=∬(Qx−Py) dA.\oint P\,dx+Q\,dy=\iint(Q_x-P_y)\,dA.

Watch for

Keep the region on the left and check smoothness throughout its interior.

04

Divergence and Curl

2 reference blocks

Read lesson ↗

Core rule

div⁡F=Px+Qy+Rz.\operatorname{div}\mathbf F=P_x+Q_y+R_z. curl⁡F=(Ry−Qz,Pz−Rx,Qx−Py).\operatorname{curl}\mathbf F=(R_y-Q_z,P_z-R_x,Q_x-P_y).

Watch for

Divergence is scalar expansion; curl is vector rotation. Neither equals speed.

05

Surface Area and Scalar Surface Integrals

2 reference blocks

Read lesson ↗

Core rule

dS=∥ru×rv∥du dv.dS=\|\mathbf r_u\times\mathbf r_v\|du\,dv.

Watch for

Include the area scale, distinguish a wall from a closed surface, and avoid covering twice.

06

Oriented Surface Flux

2 reference blocks

Read lesson ↗

Core rule

∬SF⋅n dS=∬DF(r)⋅(ru×rv)du dv.\iint_S\mathbf F\cdot\mathbf n\,dS=\iint_D\mathbf F(\mathbf r)\cdot(\mathbf r_u\times\mathbf r_v)du\,dv.

Watch for

Check orientation before integrating. The cross product already includes the area scale.

07

The Divergence Theorem

2 reference blocks

Read lesson ↗

Core rule

∬∂VF⋅n dS=∭V∇⋅F dV.\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS=\iiint_V\nabla\cdot\mathbf F\,dV.

Watch for

Require a closed outward boundary and a smooth field throughout the enclosed solid.

08

Stokes’ Theorem

2 reference blocks

Read lesson ↗

Core rule

∮∂SF⋅dr=∬S(∇×F)⋅n dS.\oint_{\partial S}\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS.

Watch for

Use the right-hand rule. Integrate curl, not the original field, across the surface.

09

Calculus III: Engineering Checkpoint

2 reference blocks

Read lesson ↗

Core rule

Match the quantity to its domain: rate, curve work, surface flux or volume accumulation.

Watch for

Check orientation, Jacobians, units, smoothness and boundary conditions before choosing a theorem.