Units & Reasonableness Checks - Calculus 3

1. A physics student calculates the flux of an electric field $\vec{E} = (3x^2, 2y, z)$ N/C through a surface and obtains the result $\Phi = 15x^3 + 4y^2z + z^2$ using the divergence theorem. Which aspect of this result should immediately indicate an error?
   Answer: The result contains variables instead of being a numerical value Electric flux through a closed surface must yield a single numerical value (in units of N⋅m²/C), not an expression containing variables. The flux integral over a specific surface with defined boundaries should eliminate all spatial variables. Choice A is incorrect because flux sign depends on surface orientation. Choice C is wrong since the divergence theorem applies to any vector field. Choice D is incorrect as the dimensional analysis could be consistent if proper surface elements were included.
2. A student calculates work done by force field $\vec{F} = (xy^2, x^2y, z)$ N along curve $C$ from $(0,0,0)$ to $(1,1,1)$ and gets $W = \frac{5}{3}$. Before checking the calculation, what should raise immediate concern about this answer?
   Answer: The answer lacks proper units for a work calculation Work has units of energy (Joules = N⋅m). Since force is given in Newtons and the path has meter dimensions, the work should be expressed as $\frac{5}{3}$ J, not as a dimensionless number. Choice A is incorrect since work sign depends on force direction relative to motion. Choice B cannot be assessed without the specific path. Choice D is irrelevant to the units issue and doesn't indicate an error in the numerical result.
3. A student models the heat flow through a cylindrical metal rod using the equation $\frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2}$, where $T$ is temperature, $t$ is time, $x$ is position, and $k$ is the thermal diffusivity. If $T$ is measured in Kelvin, $t$ in seconds, and $x$ in meters, what are the correct units for $k$?
   Answer: $\text{m}^2/\text{s}$ From dimensional analysis of the heat equation: $[\frac{\partial T}{\partial t}] = [\text{K}/\text{s}]$ and $[\frac{\partial^2 T}{\partial x^2}] = [\text{K}/\text{m}^2]$. For the equation to be dimensionally consistent, $[k][\text{K}/\text{m}^2] = [\text{K}/\text{s}]$, so $[k] = [\text{m}^2/\text{s}]$. Choice B incorrectly includes temperature units. Choice C omits spatial dimensions. Choice D has incorrect time dimension.
4. A student models the electric potential using $V(x,y,z) = \frac{kQ}{\sqrt{x^2 + y^2 + z^2}}$ and calculates the electric field as $\vec{E} = -\nabla V$. If $Q = 2 \times 10^{-6}$ C and $k = 9 \times 10^9$ N⋅m²/C², they find $|\vec{E}|$ at distance $r = 0.1$ m to be $1.8 \times 10^6$. What should make them question this result?
   Answer: The result lacks units, which should be N/C or V/m Electric field strength must have units of N/C (or equivalently V/m). The calculation $|\vec{E}| = \frac{kQ}{r^2} = \frac{(9 \times 10^9)(2 \times 10^{-6})}{(0.1)^2} = 1.8 \times 10^6$ N/C is numerically correct but lacks units. Choice A is wrong since $1.8 \times 10^6$ N/C is reasonable for this charge and distance. Choice B is incorrect as $\vec{E} = -\nabla V$ is the correct relationship. Choice D misunderstands the radial distance concept.
5. A student uses Green's theorem to find the area of region $R$ enclosed by curve $C$ using $\text{Area} = \oint_C x \, dy$ and obtains the result 25 m. However, when checking by calculating $\iint_R dA$ directly, they get 25 m². Which statement best explains this discrepancy?
   Answer: Both calculations give the same area value, but the line integral result was reported with incorrect units Both $\oint_C x \, dy$ and $\iint_R dA$ correctly calculate the area and should give the same result: 25 m². The discrepancy is that the line integral result was incorrectly reported as "25 m" instead of "25 m²". Area must have units of length squared. Choice A is wrong if $C$ is simple and closed. Choice B is incorrect since $\oint_C x \, dy$ does give area by Green's theorem. Choice C is wrong as $\iint_R dA$ is the correct area formula.
6. A student calculates the moment of inertia of a solid about the z-axis using $I_z = \iiint_D (x^2 + y^2) \rho(x,y,z) \, dV$ where $\rho = 8000$ kg/m³. After integration over a cylindrical region of radius 0.2 m and height 0.5 m, they obtain $I_z = 0.628$ kg⋅m². A quick estimate using $I \approx MR^2$ gives a different result. What suggests an error?
   Answer: The cylindrical coordinates should give $I_z \approx MR^2/2 \approx 0.4$ kg⋅m², indicating the result is too large For a solid cylinder about its central axis, $I_z = \frac{1}{2}MR^2$. The mass is $M = \rho V = 8000 \times \pi(0.2)^2(0.5) \approx 20.1$ kg, so $I_z \approx \frac{1}{2}(20.1)(0.04) \approx 0.4$ kg⋅m². The calculated 0.628 kg⋅m² is about 50% too large. Choice A is wrong since 8000 kg/m³ is reasonable for metals. Choice B is incorrect for rotation about z-axis. Choice C uses $MR^2$ instead of $MR^2/2$ for a solid cylinder.
7. An environmental science student models pollutant concentration using $c(x,y,t) = 50e^{-0.1t}\cos(\pi x/L)\sin(\pi y/L)$ mg/L, where $x$ and $y$ are in kilometers and $t$ is in hours. The student then calculates $\frac{\partial c}{\partial x} = -\frac{50\pi}{L}e^{-0.1t}\sin(\pi x/L)\sin(\pi y/L)$ and claims this represents the "concentration gradient." What unit inconsistency exists?
   Answer: The partial derivative has units mg/(L⋅km), not a proper concentration gradient When analyzing multivariable functions in applied contexts, always check dimensional consistency—the units must make physical sense throughout your calculations. The concentration function $c(x,y,t)$ has units of mg/L. When you take $\frac{\partial c}{\partial x}$, you're finding how concentration changes with respect to distance. Since $x$ is in kilometers, this partial derivative represents the rate of change of concentration per unit distance, giving units of (mg/L)/km = mg/(L⋅km). However, a true concentration gradient should describe how much the concentration changes per unit distance, typically expressed in mg/L per km, which would be written as (mg/L)/km but understood as mg/(L⋅km). The student's calculation is mathematically correct, but calling it simply a "concentration gradient" without acknowledging the proper units creates confusion about what the quantity actually represents. Looking at the wrong answers: B) is incorrect because exponential and trigonometric functions are dimensionless—the exponential $e^{-0.1t}$ and trigonometric terms don't need matching units. C) is wrong because $L$ represents a spatial length scale (in km) for the pollution pattern, not time. D) misunderstands the concept—concentration gradients are indeed first derivatives (like $\nabla c$), while second derivatives would give information about curvature or diffusion rates. Study tip: Always perform dimensional analysis when working with applied multivariable functions. Write out the units at each step of differentiation to catch inconsistencies and ensure your final answer makes physical sense.
8. A student solves for the center of mass of a solid region $D$ and obtains $\bar{x} = \frac{\iiint_D x \rho(x,y,z) \, dV}{\iiint_D \rho(x,y,z) \, dV} = 2.5 \text{ kg}$. What indicates this result is unreasonable?
   Answer: The coordinate of center of mass has units of mass instead of length The center of mass coordinate $\bar{x}$ represents a position and must have units of length (meters), not mass (kg). The numerator has units of kg⋅m and denominator has units kg, giving m. Choice A is false since non-uniform density shifts the center of mass. Choice C is incorrect as the formula works for variable density. Choice D is irrelevant to the units issue and coordinate choice doesn't affect the final result.

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