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# Integrals for Mass Calculations * **Mass of a solid with variable density**: * $m = \iiint \rho(x,y,z) \,dV$ (Triple integral over the volume) # Vector Fields * **Function f(x, y, z):** assigns a vector to each point. * **Gradient field:** a vector field derived from a scalar potential functi...

# Integrals for Mass Calculations * **Mass of a solid with variable density**: * $m = \iiint \rho(x,y,z) \,dV$ (Triple integral over the volume) # Vector Fields * **Function f(x, y, z):** assigns a vector to each point. * **Gradient field:** a vector field derived from a scalar potential function. * **Flow lines:** tangent to the vector field; curves perpendicular to the gradient of ∅. # Line Integrals * **Line integral of a scalar field:** $\int_C f(x,y,z)\,ds$ * **Line integral of a vector field:** $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C \mathbf{F} \cdot \mathbf{T} \,ds$ * Where: * $C$ is the curve * $\mathbf{F}$ is the vector field * $\mathbf{T}$ is the unit tangent vector to the curve. * $ds$ is the arc length element. # Conservative Vector Fields * **Conservative vector field**: The fundamental theorem states that * $\int_C \mathbf{F} \cdot d\mathbf{r} = \phi(B) - \phi(A)$ * if $\mathbf{F} = \nabla\phi$ (where $\phi$ is a scalar potential function). # Green's Theorem * Relates a line integral over a closed curve C to a double integral over the region R enclosed by C. * $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$ * Note that $\mathbf{F} = P\mathbf{i} + Q\mathbf{j}$ (vector field). * Use Green's theorem for: * Area of a region: $\int_C f \,dx \,dy$ or $\int_C f \,dy \,dx$ * Circulation and flux → curl → divergence