Vector calculus takes us beyond basic derivatives and integrals. It's all about working with quantities that have both size and direction in multiple dimensions. This powerful math lets us analyze complex systems in physics and engineering.

We'll learn about vector operations, line and surface integrals, and key theorems that connect different types of integrals. We'll also see how to apply these concepts in various coordinate systems, opening up new ways to solve real-world problems.

Vector operations and geometric interpretations

Vector fundamentals and arithmetic

• Vectors are mathematical objects with both magnitude and direction, represented by directed line segments or ordered pairs/triples of real numbers (e.g., $\vec{v} = \langle 1, 2, 3 \rangle$)
• Vector addition follows the parallelogram law or triangle law, with the resultant vector found by placing the initial point of one vector at the terminal point of the other ($\vec{a} + \vec{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle$)
  • Geometrically, this corresponds to placing the vectors head-to-tail and drawing the resultant from the tail of the first vector to the head of the last vector
• Vector subtraction involves adding the negative of the subtrahend ($\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$)
• Scalar multiplication changes the magnitude of a vector by the scalar factor and reverses direction if the scalar is negative ($c\vec{v} = \langle cv_1, cv_2, cv_3 \rangle$)

Dot product and cross product

• The dot product of two vectors is a scalar value, equal to the product of the magnitudes of the vectors and the cosine of the angle between them ($\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos(\theta)$)
  • It represents the projection of one vector onto another and can be used to determine the angle between vectors
  • In component form, the dot product is computed by multiplying corresponding components and summing the results ($\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$)
• The cross product of two vectors is a vector perpendicular to both original vectors, with magnitude equal to the product of the magnitudes of the vectors and the sine of the angle between them ($|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin(\theta)$)
  • Its direction is determined by the right-hand rule (curl the fingers of your right hand from the first vector to the second, and your thumb points in the direction of the cross product)
  • In component form, the cross product is computed using the determinant of a 3x3 matrix ($\vec{a} \times \vec{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$)

Line and surface integrals

Line integrals

• Line integrals measure the accumulation of a scalar or vector quantity along a curve in space
• For scalar functions, the line integral is the integral of the function along the curve, representing the area under the curve when the function is plotted along the path ($\int_C f(x, y, z) ds$)
  • The line integral is computed by parameterizing the curve and integrating with respect to the parameter ($\int_a^b f(\vec{r}(t)) |\vec{r}'(t)| dt$)
• For vector fields, the line integral can be thought of as the work done by the field along the curve ($\int_C \vec{F} \cdot d\vec{r}$)
  • It is computed by integrating the dot product of the vector field with the unit tangent vector of the curve ($\int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt$)

Surface integrals

• Surface integrals measure the accumulation of a scalar or vector quantity over a surface in space
• For scalar functions, the surface integral is the double integral of the function over the given surface, often used to find the area of a curved surface ($\iint_S f(x, y, z) dS$)
  • The surface integral is computed by parameterizing the surface and integrating with respect to the parameters ($\iint_D f(\vec{r}(u, v)) |\vec{r}_u \times \vec{r}_v| du dv$)
• For vector fields, the surface integral can represent the flux of the field through the surface ($\iint_S \vec{F} \cdot d\vec{S}$)
  • It is computed by integrating the dot product of the vector field with the unit normal vector to the surface ($\iint_D \vec{F}(\vec{r}(u, v)) \cdot (\vec{r}_u \times \vec{r}_v) du dv$)

Green's, Stokes', and Divergence Theorems

Green's and Stokes' Theorems

• Green's Theorem relates a line integral around a simple closed curve to a double integral over the region bounded by the curve ($\oint_C \vec{F} \cdot d\vec{r} = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA$)
  • It is useful for computing the work done by a vector field along a closed path
  • The theorem equates the circulation of a vector field around a closed curve to the flux of the curl of the field through the enclosed surface
• Stokes' Theorem generalizes Green's Theorem to higher dimensions, relating the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface ($\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r}$)
  • The curl of a vector field measures its rotational tendency at each point, represented as a vector perpendicular to the plane of rotation ($\nabla \times \vec{F} = \langle \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \rangle$)

Divergence Theorem

• The Divergence Theorem, also known as Gauss's Theorem, relates the surface integral of a vector field over a closed surface to the volume integral of the divergence of the field over the region enclosed by the surface ($\iint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV$)
  • The divergence of a vector field measures its tendency to emanate from or converge towards a point, represented as a scalar value ($\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$)
  • The theorem equates the flux of a vector field through a closed surface to the total divergence within the enclosed volume
• These theorems provide powerful tools for converting between line, surface, and volume integrals, simplifying many vector calculus computations
  • They allow for the evaluation of integrals over complex domains by transforming them into integrals over simpler domains (e.g., converting a surface integral to a line integral along the boundary)

Vector calculus in coordinate systems

Cartesian and cylindrical coordinates

• Vector calculus can be applied in different coordinate systems, such as Cartesian (rectangular), cylindrical, and spherical coordinates
• In Cartesian coordinates, points are represented by ordered triples $(x, y, z)$, and vectors are expressed in terms of their components along the standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$
  • Gradients, divergences, and curls are computed using partial derivatives with respect to $x$, $y$, and $z$
• Cylindrical coordinates $(r, \theta, z)$ are useful for problems with cylindrical symmetry, where $r$ is the radial distance, $\theta$ is the azimuthal angle, and $z$ is the vertical height
  • The position vector is represented as $\vec{r} = r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j} + z \hat{k}$
  • The scale factors for cylindrical coordinates are $h_r = 1$, $h_\theta = r$, and $h_z = 1$, which are used to compute arc lengths, surface areas, and volume elements

Spherical coordinates

• Spherical coordinates $(\rho, \theta, \phi)$ are useful for problems with spherical symmetry, where $\rho$ is the radial distance, $\theta$ is the azimuthal angle, and $\phi$ is the polar angle
  • The position vector is represented as $\vec{r} = \rho \sin(\phi) \cos(\theta) \hat{i} + \rho \sin(\phi) \sin(\theta) \hat{j} + \rho \cos(\phi) \hat{k}$
  • The scale factors for spherical coordinates are $h_\rho = 1$, $h_\theta = \rho \sin(\phi)$, and $h_\phi = \rho$
• When working in different coordinate systems, it is essential to use the appropriate unit vectors and scale factors for computing gradients, divergences, and curls
  • Transforming between coordinate systems involves substituting the appropriate expressions for the coordinates and scale factors and simplifying the resulting expressions
  • For example, the gradient in spherical coordinates is given by $\nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho \sin(\phi)} \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{\rho} \frac{\partial f}{\partial \phi} \hat{\phi}$