Multivariable and Vector Calculus

Introduction

Electronic and communication systems are governed by quantities that vary with several independent variables. Electric potential may depend on three spatial coordinates and time; temperature in a semiconductor depends on position; antenna radiation varies with direction; and electromagnetic energy flows through surfaces. Multivariable calculus provides the language for describing such behavior, while vector calculus connects local field variation to global measurements such as circulation and flux.

This chapter develops partial differentiation, gradients, Jacobians, multiple integration, moments, coordinate transformations, vector fields, divergence, curl, and line and surface integrals. Green’s, Gauss’s, and Stokes’ theorems are then used to connect differential and integral field models. Applications emphasize electrostatics, electromagnetic induction, charge conservation, transmission structures, and computational field analysis.

Learning Objectives

After completing this chapter, students should be able to:

1. Compute and interpret partial derivatives and total differentials.
2. Apply the chain rule, Taylor approximation, and tests for multivariable extrema.
3. Calculate gradients and directional derivatives.
4. Form and interpret Jacobian matrices and determinants.
5. Evaluate double and triple integrals over engineering regions.
6. Calculate mass, centroids, first moments, and second moments.
7. Transform integrals between Cartesian, cylindrical, and spherical coordinates.
8. Compute line integrals, surface integrals, divergence, and curl.
9. Apply Green’s, Gauss’s, and Stokes’ theorems under their proper conditions.
10. Use vector calculus in electrostatic and electromagnetic field models.

1. Functions of Several Variables

A scalar-valued function of several variables assigns one scalar to each point in its domain. Examples include electric potential φ(x,y,z), temperature T(x,y,z,t), and charge density ρ_v(x,y,z). A vector-valued function assigns a vector to each point, as in the electric field E(x,y,z) or magnetic flux density B(x,y,z,t).

A limit of f(x,y) at (a,b) exists only if the same value is approached along every possible path. Continuity at that point requires

Checking a few paths can disprove a limit but generally cannot prove that it exists. A proof normally requires an inequality, coordinate transformation, or continuity theorem.

2. Partial Derivatives and Local Approximation

2.1 Partial derivatives

The partial derivative with respect to one variable measures change while the other independent variables are held constant:

Higher derivatives include f_xx, f_yy, and the mixed derivatives f_xy and f_yx. If the second partial derivatives are continuous near a point, Clairaut’s theorem gives

This equality is not guaranteed when the required continuity conditions fail.

Worked Example 1: Partial derivatives with explicit units

Let v be voltage measured in volts (V), and let Θ be absolute temperature measured in kelvins (K). Consider the model

where k=3 V². Both terms therefore have units V²·K, so they can be added dimensionally.

The partial derivatives are

At v=2 V and Θ=300 K,

and

The derivative units follow from output units divided by the units of the differentiated variable.

2.2 Total differential and linearization

If z=f(x,y) is differentiable, then a small output change is approximated by

At (a,b), the tangent-plane approximation is

The approximation is reliable sufficiently close to the expansion point when the function is differentiable.

2.3 Chain rule

If z=f(x,y), where x=x(t) and y=y(t), then

More generally, if x=x(u,v) and y=y(u,v), then

2.4 Taylor expansion and stationary points

A second-order Taylor approximation about (a,b) is

where all derivatives on the right are evaluated at (a,b).

At an interior stationary point, f_x=f_y=0. Define

If D>0 and f_xx>0, the point is a local minimum. If D>0 and f_xx<0, it is a local maximum. If D<0, it is a saddle point. When D=0, the test is inconclusive.

For constrained optimization with g(x,y,z)=c, Lagrange multipliers require

provided the constraint is sufficiently smooth and ∇g≠0 at the candidate point.

3. Gradient and Directional Derivatives

For a differentiable scalar field f(x,y,z), its gradient is

where a_x, a_y, and a_z are Cartesian unit vectors. The gradient points in the direction of greatest local increase. Its magnitude gives the maximum directional rate of change.

For a unit vector u, the directional derivative is

The vector must be normalized. If a direction is given by a nonzero vector d, use u=d/|d|.

Worked Example 2: Electric potential and field

Let the electric potential be

where numerical coefficients carry the units needed when x, y, and z are measured in metres. The electrostatic field is

At (x,y,z)=(0.10,0.20,0.30) m,

The negative sign means that the electric field points toward decreasing potential.

4. Jacobians and Coordinate Transformations

4.1 Jacobian matrix

For a transformation u=u(x,y), v=v(x,y), the Jacobian matrix is

Its determinant is

A nonzero determinant indicates local invertibility under the inverse function theorem. The determinant also measures local area scaling, including orientation. Integration uses its absolute value.

If x=x(u,v) and y=y(u,v), then

Figure: A Jacobian maps a small parameter-space area into a scaled physical-space area.

4.2 Standard coordinate systems

For polar coordinates,

For cylindrical coordinates,

For spherical coordinates, using polar angle θ from the positive z-axis and azimuth φ in the xy-plane,

The factor r, or r² sinθ, must not be omitted. Coordinate limits must describe the region exactly and without unintended multiple coverage.

5. Multiple Integrals, Mass, and Moments

5.1 Double and triple integrals

A double integral accumulates a quantity over an area:

If f=1, the result is the area of R. A triple integral accumulates over a volume:

Fubini’s theorem permits repeated integration when the function satisfies appropriate integrability conditions; continuity on a bounded region is a common sufficient condition.

5.2 Mass and moments

For a thin lamina with surface mass density ρ(x,y) in kg/m²,

The first moments about the coordinate axes are

Their units are kg·m. The centroid is

The mass moments of inertia about the axes are

These have units kg·m². For geometric second moments of area, omit ρ; the units then become m⁴. Mass moments and area moments must not be confused.

Worked Example 3: Mass, first moments, and centroid

A rectangular lamina occupies 0≤x≤a, 0≤y≤b and has density

where ρ₀ is a constant surface density. Its mass is

The first moment about the y-axis is

The first moment about the x-axis is

Therefore,

The density increases with x, so the result x̄>a/2 is physically reasonable.

Worked Example 4: Volume in cylindrical coordinates

For a cylinder 0≤r≤R, 0≤φ≤2π, and 0≤z≤h,

The factor r is the cylindrical-coordinate Jacobian.

6. Vector Fields and Differential Operators

A vector field in Cartesian coordinates is

6.1 Divergence

The divergence measures net outward source strength per unit volume:

For F=xa_x+ya_y+za_z, ∇·F=3. Positive divergence indicates a local source; negative divergence indicates a local sink. Zero divergence does not imply that the field itself is zero.

6.2 Curl

The curl measures local circulation density:

If F=∇φ and φ has continuous second derivatives, then

Conversely, a continuously differentiable curl-free field has a scalar potential on a simply connected domain. Domain topology matters: a region containing a hole may support curl-free fields with nonzero circulation.

Another important identity is

provided the required second derivatives are continuous.

In cylindrical coordinates, for F=F_ra_r+F_φa_φ+F_za_z,

Coordinate-specific formulas include scale factors and should not be replaced by Cartesian formulas.

7. Line and Surface Integrals

7.1 Scalar and vector line integrals

For a curve C parameterized by r(t), a≤t≤b,

The circulation or work integral of a vector field is

Reversing curve orientation changes the sign of a vector line integral. If F=∇φ on a suitable domain, then the fundamental theorem for line integrals gives

so the result is independent of path.

7.2 Surface integrals and flux

For a parameterized surface r(u,v), an oriented vector area element is

The flux of F through the oriented surface is

Changing orientation reverses the sign. For a constant field normal to a planar area A, the special-case formula is Φ=FA. In general, the dot product and integration cannot be omitted.

8. Green’s, Gauss’s, and Stokes’ Theorems

These theorems connect local derivatives with global integrals.

8.1 Green’s theorem

Let C be the positively oriented, simple, closed, piecewise-smooth boundary of a planar region R. If P and Q have continuous first partial derivatives on an open region containing R, then

Its flux form is

Green’s theorem can replace a difficult boundary integral by an easier area integral, or vice versa.

8.2 Gauss’s divergence theorem

Let V be a bounded solid with closed, piecewise-smooth boundary S, oriented outward. If F has continuous first partial derivatives on an open region containing V, then

The theorem cannot be applied directly across an unaccounted singularity. A point charge, for example, requires excluding a small region around the singularity or using a distributional formulation.

8.3 Stokes’ theorem

Let S be an oriented, piecewise-smooth surface with positively oriented boundary C. If F is continuously differentiable on a region containing S, then

The boundary orientation follows the right-hand rule relative to the chosen normal. Any surface spanning the same boundary may be used if the field is sufficiently smooth throughout the intervening region.

Theorem comparison

9. Electromagnetic Applications

Vector calculus gives Maxwell’s equations their differential and integral forms. In SI units,

Here D is electric flux density in C/m², ρ_v is volume charge density in C/m³, B is magnetic flux density in teslas, E is electric field in V/m, H is magnetic field intensity in A/m, and J is current density in A/m².

Gauss’s theorem transforms ∇·D=ρ_v into

Stokes’ theorem transforms Faraday’s law into

for a fixed surface and contour. Moving circuits require additional motional-emf considerations.

Taking the divergence of the Ampère–Maxwell equation and using ∇·(∇×H)=0 gives

which is the differential continuity equation expressing charge conservation.

Worked Example 5: Magnetic flux

A uniform magnetic field B=0.5 T is normal to a circular loop of radius r=0.020 m. The magnetic flux is

This use of BA is valid because the field is uniform and perpendicular to the loop.

10. Applied Case: Coaxial Cable Field and Capacitance

Background

A coaxial cable contains an inner cylindrical conductor of radius a and an outer conductor whose inner radius is b. A homogeneous dielectric of permittivity ε fills the region a<r<b. This is an illustrative idealized case: the cable is assumed sufficiently long that end effects are negligible.

Problem

For a=1.0 mm, b=4.0 mm, relative permittivity ε_r=2.25, and potential difference V₀=100 V, determine the capacitance per metre, line charge density, and maximum electric-field magnitude.

Analysis and approach

Cylindrical symmetry makes the electric field radial and dependent only on r. Apply Gauss’s law to a coaxial Gaussian cylinder of radius r and length L:

Hence

The potential difference is

Therefore,

With ε=ε₀ε_r=(8.854×10⁻¹²)(2.25) F/m,

The line charge density is

The field is greatest at the smallest radius:

Outcome

The model predicts a capacitance of approximately 90.3 pF/m and a maximum dielectric field of 72.1 kV/m. A practical design must compare this field with dielectric strength while accounting for conductor roughness, manufacturing tolerances, connectors, temperature, and transient overvoltage.

Learning points

1. Symmetry determines an efficient coordinate system and Gaussian surface.
2. Gauss’s theorem converts a field problem into an algebraic relation.
3. Potential follows from a line integral of the electric field.
4. The ideal infinite-cable model becomes less accurate near terminations.

11. Computational Practice, Limitations, and Professional Responsibility

Real field geometries often lack closed-form solutions. Finite-element and finite-difference software approximates derivatives and integrals on meshes. Symbolic algebra and AI-assisted systems can help construct Jacobians, differentiate fields, or propose coordinate transformations, but their outputs require independent checks.

Good professional practice includes:

• checking dimensions and SI units;
• drawing the domain and specifying all orientations;
• testing whether singularities violate theorem assumptions;
• validating numerical results against symmetry, limiting cases, and conservation laws;
• conducting mesh-convergence studies;
• reporting material assumptions and boundary conditions;
• retaining adequate precision until the final result.

Field predictions affect insulation safety, electromagnetic exposure assessment, antenna compliance, medical electronics, and interference control. Engineers must use validated material data and applicable standards rather than adjusting models to produce convenient outcomes. Numerical visualizations should disclose scale choices because color maps can exaggerate or conceal field concentrations.

Contemporary electromagnetic design increasingly uses automatic differentiation, adaptive meshing, reduced-order models, and physics-informed machine learning. These methods improve speed but do not remove the need to enforce Maxwell’s equations, boundary conditions, charge conservation, and uncertainty analysis.

Chapter Summary

Multivariable calculus extends differentiation and integration to quantities depending on several variables. Partial derivatives describe coordinate-wise rates of change, while total differentials and Taylor expansions provide local approximations. The gradient identifies maximum increase and produces directional derivatives. Jacobians describe local transformation and supply the required area or volume scaling in changed coordinates.

Double and triple integrals calculate area, volume, mass, charge, and accumulated field quantities. First moments determine centroids, while second moments describe rotational mass distribution or geometric resistance to bending. Vector fields are analyzed through divergence and curl. Line integrals measure work or circulation, and surface integrals measure flux.

Green’s, Gauss’s, and Stokes’ theorems relate boundary measurements to derivatives over interiors, subject to smoothness, orientation, and domain requirements. In electromagnetics, these results connect Maxwell’s differential equations to their integral forms and support practical analysis of cables, capacitors, antennas, induction, and charge conservation.

Key Terms

Partial derivative: Rate of change with one independent variable while others are held constant.

Total differential: First-order approximation of the total change in a differentiable function.

Gradient: Vector of partial derivatives pointing toward greatest local increase.

Directional derivative: Rate of scalar-field change in a specified unit-vector direction.

Jacobian matrix: Matrix containing the first derivatives of a multivariable transformation.

Jacobian determinant: Signed local area or volume scaling factor of a transformation.

Double integral: Accumulation of a scalar quantity over a planar region.

Triple integral: Accumulation of a scalar quantity over a volume.

First moment: Density-weighted distance integral used to locate a centroid.

Moment of inertia: Second moment measuring mass distribution relative to an axis or point.

Vector field: Assignment of a vector to every point in a domain.

Divergence: Scalar measure of local net outward flux density.

Curl: Vector measure of local circulation density.

Line integral: Integral of a scalar or vector field along a curve.

Flux: Net passage of a vector field through an oriented surface.

Conservative field: Vector field representable as the gradient of a scalar potential.

Green’s theorem: Relation between a closed planar boundary integral and an area integral.

Divergence theorem: Relation between outward flux through a closed surface and volume divergence.

Stokes’ theorem: Relation between circulation around a boundary and curl flux through a surface.

Review Questions

Short Answer Questions

1. What physical and geometric information is conveyed by the gradient?
2. Why is the absolute value of a Jacobian determinant used in transformed integrals?
3. Distinguish between divergence and curl.
4. State the relationship between first moments and the centroid of a lamina.
5. Why must orientation be specified for a flux integral?

Long Answer Questions

1. Derive the second-order test for classifying stationary points using the Hessian quadratic form.
2. Explain polar, cylindrical, and spherical coordinate transformations, including their Jacobian factors.
3. Develop the formulas for mass, centroid, and mass moments of inertia of a variable-density lamina.
4. State and explain Green’s, Gauss’s, and Stokes’ theorems, including their applicability conditions.
5. Show how Gauss’s and Stokes’ theorems connect the differential and integral forms of Maxwell’s equations.

Analytical / Application Questions

1. A dielectric cylinder has volume charge density ρ_v(r)=ρ₀(1−r²/R²). Use cylindrical coordinates to calculate the total charge in a cylinder of radius R and length L.
2. Determine whether F=(2xy+z)a_x+x²a_y+xa_z is conservative on R³. If so, find a scalar potential.
3. For a coaxial cable with a radially varying permittivity ε(r), formulate the integrals required to determine electric field, voltage, and capacitance per unit length.

Multiple Choice Questions with Answers

Question 1

For f(x,y)=x²y+3y, what is ∂f/∂x?

A. 2xy

B. x²+3

C. 2x+3

D. x²y

Correct Answer: A. 2xy

Question 2

If ∇f=(3,4,0), what is the maximum directional derivative of f at that point?

A. 3

B. 4

C. 5

D. 7

Correct Answer: C. 5

Question 3

What is the Jacobian area factor for the polar transformation x=r cosθ, y=r sinθ?

A. 1

B. r

C. r²

D. sinθ

Correct Answer: B. r

Question 4

A lamina of mass M has first moment Q_y about the y-axis. Its x-coordinate of centroid is:

A. M/Q_y

B. Q_y/M

C. Q_yM

D. Q_y/M²

Correct Answer: B. Q_y/M

Question 5

Which volume element is correct in spherical coordinates when θ is measured from the positive z-axis?

A. r dr dθ dφ

B. r² cosθ dr dθ dφ

C. r² sinθ dr dθ dφ

D. sinθ dr dθ dφ

Correct Answer: C. r² sinθ dr dθ dφ

Question 6

For F=xa_x+ya_y+za_z, the divergence ∇·F equals:

A. 0

B. 1

C. 2

D. 3

Correct Answer: D. 3

Question 7

On a simply connected domain, a continuously differentiable vector field with zero curl is necessarily:

A. Solenoidal only

B. Conservative

C. Spatially constant

D. Of zero divergence

Correct Answer: B. Conservative

Question 8

A uniform magnetic field of 0.5 T is normal to a circular loop of radius 0.020 m. What is the magnetic flux?

A. 0.0002π Wb

B. 0.0004π Wb

C. 0.001π Wb

D. 0.02π Wb

Correct Answer: A. 0.0002π Wb

Question 9

Which theorem directly converts the circulation around a closed space curve into a surface integral of curl?

A. Fubini’s theorem

B. Gauss’s theorem

C. Stokes’ theorem

D. Taylor’s theorem

Correct Answer: C. Stokes’ theorem

Question 10

For an ideal coaxial cable with homogeneous permittivity ε and conductor radii a<b, the capacitance per unit length is:

A. 2πε ln(b/a)

B. 2πε/ln(b/a)

C. ε(b−a)

D. πε(b²−a²)

Correct Answer: B. 2πε/ln(b/a)