Multivariable & Vector Calculus

A vector $\mathbf{v} = \langle a, b, c \rangle$ in $\mathbb{R}^3$ has magnitude $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$ and direction.

Dot product: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\mathbf{u}||\mathbf{v}|\cos\theta$. It measures how aligned two vectors are. Zero dot product means perpendicular.

Cross product: $\mathbf{u} \times \mathbf{v}$ is a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, with magnitude $|\mathbf{u}||\mathbf{v}|\sin\theta$ (the area of the parallelogram they span).

Equations of lines: $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$ (point plus direction). Equations of planes: $\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$ (normal vector dotted with displacement is zero).

These tools let you describe points, lines, planes, and curves in 3D space — the setting for everything that follows.

A vector function $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$ traces a curve in 3D space as the parameter $t$ varies. Think of $t$ as time: at each moment, the point $(x(t), y(t), z(t))$ is the particle's position.

The derivative $\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$ is the velocity vector — it points in the direction of motion and its magnitude is the speed.

Speed: $|\mathbf{r}'(t)| = \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}$. The unit tangent vector $\mathbf{T}(t) = \mathbf{r}'(t)/|\mathbf{r}'(t)|$ gives the direction of motion without the speed information.

Arc length: $L = \int_a^b |\mathbf{r}'(t)|\,dt$ — integrate speed over time to get total distance traveled. This is the 3D generalization of $L = \int \sqrt{1+[f']^2}\,dx$.

Curvature $\kappa$ measures how sharply the curve turns: $\kappa = |\mathbf{T}'(t)|/|\mathbf{r}'(t)| = |\mathbf{r}' \times \mathbf{r}''|/|\mathbf{r}'|^3$. A straight line has $\kappa = 0$; a circle of radius $r$ has $\kappa = 1/r$ (tighter circle = higher curvature).

The acceleration vector $\mathbf{a} = \mathbf{r}''(t)$ naturally decomposes into two components: $\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}$ where $a_T = \frac{d}{dt}|\mathbf{r}'|$ (tangential — speeding up or slowing down) and $a_N = \kappa|\mathbf{r}'|^2$ (normal — turning). This is why you feel pushed sideways on a curve and pressed back under acceleration.

For $f(x, y)$, the partial derivative $f_x = \frac{\partial f}{\partial x}$ differentiates with respect to $x$ while treating $y$ as a constant.

Geometrically: $f_x(a, b)$ is the slope of the surface $z = f(x, y)$ in the $x$-direction at the point $(a, b)$.

Higher-order partials: $f_{xx}$, $f_{yy}$, $f_{xy}$, $f_{yx}$. Clairaut's Theorem: if $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$ (the order of differentiation doesn't matter).

Example: if $f(x,y) = x^2 y + \sin(xy)$, then $f_x = 2xy + y\cos(xy)$ and $f_y = x^2 + x\cos(xy)$.

Limits and continuity in multiple variables: $\lim_{(x,y)\to(a,b)} f(x,y) = L$ means $f$ approaches $L$ from every possible path. To show a limit doesn't exist, find two paths that give different limits.

The gradient of $f(x,y)$ is the vector $\nabla f = \langle f_x, f_y \rangle$. It encodes all the directional information about how $f$ changes. Its direction points toward the steepest uphill, and its magnitude $|\nabla f|$ is the rate of steepest ascent.

The directional derivative of $f$ in the direction of unit vector $\mathbf{u}$ is $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = |\nabla f|\cos\theta$, where $\theta$ is the angle between $\nabla f$ and $\mathbf{u}$. This tells you the rate of change of $f$ in any direction.

Key consequences: $D_{\mathbf{u}} f$ is maximized when $\mathbf{u}$ points in the direction of $\nabla f$ (steepest ascent, $\theta=0$). It's minimized in the direction of $-\nabla f$ (steepest descent, $\theta=\pi$). It's zero when $\mathbf{u}$ is perpendicular to $\nabla f$ — this is the 'contour direction' where $f$ stays constant.

The gradient is perpendicular to level curves (in 2D) and level surfaces (in 3D). This is why contour maps work: if you walk along a contour line ($f =$ constant), you're moving perpendicular to $\nabla f$, so $f$ doesn't change.

The tangent plane to $z = f(x,y)$ at $(a, b, f(a,b))$ is $z - f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b)$. This is the multivariable analog of the tangent line — the best linear approximation to the surface at that point.

To find extrema of $f(x,y)$: set $f_x = 0$ and $f_y = 0$ simultaneously. Solutions are critical points.

Second derivative test: compute $D = f_{xx} f_{yy} - (f_{xy})^2$ at a critical point. If $D > 0$ and $f_{xx} > 0$: local min. If $D > 0$ and $f_{xx} < 0$: local max. If $D < 0$: saddle point. If $D = 0$: inconclusive.

Lagrange multipliers solve constrained optimization: maximize/minimize $f(x,y)$ subject to a constraint $g(x,y) = c$.

Method: solve $\nabla f = \lambda \nabla g$ and $g(x,y) = c$ simultaneously. The scalar $\lambda$ is the Lagrange multiplier.

Geometric insight: at a constrained extremum, the gradient of $f$ must be parallel to the gradient of $g$. They point in the same (or opposite) direction along the constraint curve.

A double integral $\iint_R f(x,y)\,dA$ computes the volume under $z = f(x,y)$ over a region $R$ in the $xy$-plane.

Iterated integrals: $\iint_R f\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx$. Evaluate the inner integral first (treating the outer variable as a constant).

Fubini's Theorem: if $f$ is continuous on $R$, you can integrate in either order: $\int\int f\,dy\,dx = \int\int f\,dx\,dy$. Sometimes one order is much easier.

Polar coordinates: for circular or radial regions, use $x = r\cos\theta$, $y = r\sin\theta$, $dA = r\,dr\,d\theta$. The extra factor of $r$ is critical.

Triple integrals extend to 3D: $\iiint_E f\,dV$ computes mass (if $f$ is density), or volume (if $f = 1$).

Cylindrical coordinates $(r, \theta, z)$: $dV = r\,dr\,d\theta\,dz$. Use for problems with cylindrical symmetry.

Spherical coordinates $(\rho, \phi, \theta)$: $dV = \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$. Use for spheres and cones.

Applications: mass = $\iint \rho\,dA$, center of mass = $\frac{\iint x\rho\,dA}{\text{mass}}$, moments of inertia.

A vector field assigns a vector to every point in space: $\mathbf{F}(x,y) = \langle P(x,y), Q(x,y) \rangle$ in 2D, or $\mathbf{F}(x,y,z) = \langle P, Q, R \rangle$ in 3D. Visually, draw a small arrow at each point showing the direction and magnitude.

Physical examples: gravitational fields (arrows point toward massive objects), electric fields (arrows show the force on a positive charge), fluid velocity fields (arrows show speed and direction of flow at each point), wind maps, and magnetic fields.

A vector field is conservative if $\mathbf{F} = \nabla f$ for some scalar function $f$ (called the potential function). This means the field is the gradient of something. Conservative fields have a crucial property: the line integral between two points is independent of the path taken — only the endpoints matter.

Test for conservative fields in 2D: $\mathbf{F} = \langle P, Q \rangle$ is conservative if and only if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ on a simply connected domain (no holes). This is the 'cross-partial' test.

Finding the potential function: if $\mathbf{F} = \langle P, Q \rangle$ passes the cross-partial test, find $f$ by integrating: $f(x,y) = \int P\,dx$ (treating $y$ as constant), then determine any leftover function of $y$ by matching $f_y = Q$.

Non-conservative fields include anything with a nonzero curl (in 3D) or failing the cross-partial test. The work done by such a field depends on the path, not just the endpoints. Example: $\mathbf{F} = \langle -y, x \rangle$ has $P_y = -1$ but $Q_x = 1$, so it's not conservative — it swirls.

A line integral computes a quantity accumulated along a curve $C$ in space. There are two main types, and they answer different questions.

Scalar line integral: $\int_C f\,ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\,dt$. Here $ds = |\mathbf{r}'(t)|\,dt$ is the arc length element. This computes the 'weighted length' of the curve — for example, the mass of a wire with density $f$ at each point, or the total cost of a path where $f$ is the cost per unit length.

Vector line integral (work integral): $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt$. The dot product picks out the component of $\mathbf{F}$ in the direction of motion. This computes the work done by the force field $\mathbf{F}$ on a particle moving along $C$. Force components perpendicular to the path contribute zero work.

Fundamental Theorem for Line Integrals: if $\mathbf{F} = \nabla f$ (conservative), then $\int_C \nabla f \cdot d\mathbf{r} = f(\text{end}) - f(\text{start})$. The integral depends only on the endpoints, not the path. This is the multivariable analog of $\int_a^b F'(x)\,dx = F(b) - F(a)$.

Corollary: for a conservative field, the line integral around any closed loop is zero ($\oint_C \nabla f \cdot d\mathbf{r} = 0$). You end where you started, so the potential difference is zero.

Divergence measures how much a vector field 'spreads out' from a point: $\text{div}\,\mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$.

Positive divergence: source (field flows outward). Negative: sink (field flows inward). Zero: incompressible flow.

Curl measures the 'rotation' or 'circulation' of a field: $\text{curl}\,\mathbf{F} = \nabla \times \mathbf{F} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle$.

If $\text{curl}\,\mathbf{F} = \mathbf{0}$ everywhere, the field is irrotational (no swirling). For simply connected domains, irrotational implies conservative.

Physical examples: the curl of a fluid velocity field tells you how fast and around which axis a small paddle wheel would spin. The divergence tells you whether fluid is being created or destroyed at a point.

These three theorems are the pinnacle of vector calculus. Each relates an integral over a boundary to an integral over the region it encloses.

Green's Theorem (2D): $\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$. A line integral around a closed curve equals a double integral over the enclosed region.

Stokes' Theorem (3D surfaces): $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$. A line integral around the boundary of a surface equals the surface integral of the curl.

Divergence Theorem (3D volumes): $\oiint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\nabla \cdot \mathbf{F})\,dV$. The flux through a closed surface equals the triple integral of the divergence inside.

Unified pattern: in every case, an integral over the boundary = an integral of a derivative over the interior. This is the same idea as the Fundamental Theorem of Calculus ($\int_a^b f'(x)\,dx = f(b) - f(a)$), generalized to higher dimensions.

Green's Theorem is the 2D special case of Stokes'. The Divergence Theorem is the 3D analog for flux instead of circulation.