Vector Calculus: Cylindrical and Spherical Coordinates

A coordinate system that describes a point in 3D space using a distance from the origin, an angle from a reference axis, and a height from a reference plane.

A coordinate system that describes a point in 3D space using a distance from the origin, two angles from reference axes.

A formula used to transform a double integral from Cartesian coordinates to a new coordinate system. It involves the Jacobian determinant, which represents the scaling factor between the two coordinate systems.

A formula used to transform a triple integral from Cartesian coordinates to a new coordinate system. It involves the Jacobian determinant, which represents the scaling factor between the two coordinate systems.

The Jacobian determinant for cylindrical coordinates. It represents the scaling factor between the cylindrical and Cartesian systems, and it tells us how much the volume element changes under the coordinate transformation.

The Jacobian determinant for spherical coordinates. It represents the scaling factor between the spherical and Cartesian systems, and it tells us how much the volume element changes under the coordinate transformation.

A surface defined by the equation z² * sin² α = (x² + y²) * cos² α, where α is a constant angle.

Region in 3D space bounded above by the sphere x² + y² + z² = a² and below by the cone z² * sin² α = (x² + y²) * cos² α.

The gradient of a scalar field f is a vector field that points in the direction of the greatest rate of increase of f. It is defined as the partial derivative of f with respect to each coordinate direction.

The directional derivative of a scalar field f in the direction of a unit vector a is the rate of change of f along a line in the direction of a. It is defined as the dot product of the gradient of f and a.

A vector field F is said to be conservative if it is the gradient of a scalar potential function ϕ. This means that the line integral of F over any closed curve is zero.

A curve C in R3 is a set of points that can be parameterized by a continuous function γ: [a, b] → R3.

The tangent vector to a curve at a point γ(t) is the vector that points in the direction of the curve at that point. It is defined as the derivative of the parameterization function γ with respect to t.

The arc length of a curve C from point γ(a) to point γ(t) is the distance along the curve from the starting point to the end point. It is defined as the integral of the magnitude of the tangent vector over the interval [a, t].

The line integral of a scalar field f along a curve C is the integral of f with respect to arc length along C. It can be interpreted as the total value of f along the curve.

The line integral of a vector field F along a curve C is the integral of the dot product of F and the tangent vector to C, with respect to arc length. It can be interpreted as the work done by F along the curve.

A vector field is called solenoidal or incompressible if its divergence is zero. This means that the vector field does not expand or contract at any point.

The Jacobian matrix of a vector function $f: \mathbb{R}^m \to \mathbb{R}^n$ is a matrix containing the partial derivatives of each component of $f$ with respect to each input variable.

The divergence of a vector field $F: D \to \mathbb{R}^3$ is a scalar function that measures the expansion of the vector field at a point. It is calculated by taking the dot product of the del operator with the vector field.

The curl of a vector field $F: D \to \mathbb{R}^3$ is another vector field that measures the rotation of the original vector field at a point. It is calculated by taking the cross product of the del operator with the vector field.

The surface element $dS$ represents a small area element on a surface. It can be calculated using different formulas depending on the representation of the surface.

The unit normal vector $ hat$ at a point on a surface is a vector perpendicular to the tangent plane at that point. It can be calculated using different formulas for different surface representations.

A parametric representation of a surface describes its position using two parameters $u$ and $v$. The position of a point on the surface is determined by plugging in values for $u$ and $v$.

An implicit representation of a surface is given by an equation $F(x, y, z) = 0$. Any point that satisfies the equation lies on the surface.

The curl of a vector field measures the rotation of the vector field at a point. It is a vector quantity that points in the direction of the axis of rotation and has magnitude equal to the angular velocity of the rotation.

A vector field is irrotational if its curl is zero everywhere. This means that there is no rotation in the vector field at any point.

The gradient of the divergence of a vector field is a vector field that measures the rate of change of the divergence of the vector field.

The divergence of the curl of a vector field is always zero, regardless of the vector field.

The curl of the curl of a vector field can be expressed in terms of the gradient, divergence, and Laplacian operators. It is a vector field that represents the vector potential of the Laplacian of the curl of the vector field.

The Laplace operator is a second-order differential operator that measures the curvature of a function. It is used in many areas of physics and engineering, including electromagnetism, fluid mechanics, and heat transfer.

Laplace's equation is a second-order partial differential equation that is central to many areas of physics and engineering. It is used to model phenomena such as heat flow, electrostatic potential, and fluid flow.

If a function u is a solution to Laplace's equation, then its gradient is both solenoidal and irrotational. This means that the gradient field has no net flow out of any region (solenoidal) and no rotation (irrotational).

Green's theorem relates the line integral of a vector field around a simple closed curve to the double integral of the curl of the vector field over the region enclosed by the curve.

The Divergence Theorem is a generalization of Green's Theorem to three dimensions. It states that the flux of a vector field across a closed surface is equal to the integral of the divergence of the vector field over the volume enclosed by the surface.

Stokes' theorem relates the line integral of a vector field around a closed curve to the flux of the curl of the vector field across any surface that is bounded by the curve.

Both divergence and Stokes' theorems are fundamental theorems of vector calculus that relate integrals over volumes or surfaces to integrals over their boundaries. They are generalizations of the fundamental theorem of calculus to higher dimensions.

The divergence of a vector field measures the rate at which the vector field is expanding or contracting at a point. It's calculated by taking the dot product of the del operator and the vector field.

The flux of a vector field through a surface is the amount of the vector field that passes through the surface. It's calculated by taking the dot product of the vector field and the unit normal vector to the surface, and then integrating over the surface.

The line integral of a vector field along a curve is the work done by the vector field on a particle moving along the curve. It's calculated by taking the dot product of the vector field and the tangent vector to the curve, and then integrating over the curve.