Vectors
Term | Definition | ||
|---|---|---|---|
| Vector | An ordered set of numbers or symbols, typically arranged in a column or row. | ||
| Component | The individual parts or elements of a vector along specific axes or directions. | ||
| Vector Addition | The operation of combining two or more vectors to produce a resultant vector. | ||
| Parallelogram method | A geometric method used to add or subtract vectors graphically. It involves constructing a parallelogram using the vectors as adjacent sides, with the diagonal of the parallelogram representing the resultant vector. | ||
| Vector Subtraction | The operation of finding the difference between two vectors. This is done by subtracting the components of one vector from the corresponding components of another vector. | ||
| Scalar Multiplication | The operation of multiplying a vector by a scalar (a single numerical value). This operation scales the magnitude of the vector without changing its direction. | ||
| Dot Product | A binary operation that takes two vectors and produces a scalar. | ||
| Cross Product | A binary operation that takes two vectors and returns a vector perpendicular to the plane containing the two input vectors. The magnitude of the cross product is the product of the magnitudes of the two vectors multiplied by the sine of the angle between them. | ||
| Magnitude | The size or length of a vector, representing its distance from the origin in space. It is calculated using the Pythagorean theorem for two- and three-dimensional vectors. | ||
| Normal Vector | A vector that is perpendicular (normal) to a surface or plane at a given point. It is used to define the orientation or direction of the surface. | ||
| Unit Vector | A vector with a magnitude of 1, typically used to represent direction in space. | ||
| Zero Vector | A vector with all components equal to zero. | ||
| Equivalent Vectors | Vectors that have the same magnitude and direction, even if their initial points are different. They represent the same physical quantity or displacement. | ||
3D Space
Term | Definition | Formula | |||
|---|---|---|---|---|---|
| Line | A straight path that extends infinitely in both directions. In three-dimensional space, a line can be defined by two distinct points. | ||||
| Skew Line | Two lines in three-dimensional space that do not intersect and are not parallel. Skew lines lie in different planes and are not coplanar. | ||||
| Parallel Line | Two or more lines that lie in the same plane and never intersect. Parallel lines have the same slope and run in the same direction. | ||||
| Plane | A flat, two-dimensional surface that extends infinitely in all directions. A plane is defined by at least three non-collinear points or by a point and a normal vector. | ||||
| Scalar Equation of a Plane | An equation that represents a plane in three-dimensional space using scalar constants instead of vector or parametric equations. It is typically written in the form | ||||
| Ellipsoid | A three-dimensional geometric shape resembling a stretched sphere or oval. It is defined by three semiaxes and can be visualized as a surface where the distance from any point on the surface to three fixed points (foci) is constant. | ||||
| Elliptic Cone | A three-dimensional geometric shape formed by all lines passing through a fixed point (the vertex) and intersecting a fixed conic section (the base) at an angle. | ||||
| Elliptic Paraboloid | A three-dimensional geometric shape resembling a parabolic bowl or dish. It is formed by translating a parabola along a fixed axis, resulting in a surface with parabolic cross-sections. | ||||
| Parametric Equation of a Line | A set of equations that define the coordinates of points on a line in terms of one or more parameters. In three-dimensional space, a parametric equation of a line is often written as | ||||
| Symmetric Equations of a Line | Equations that represent a line in terms of the distances from the line to the coordinate axes. In three-dimensional space, symmetric equations of a line are typically written as | ||||
| General Equation of a Plane | An equation that represents a plane in three-dimensional space using the coordinates of points on the plane. It is typically written in the form | ||||
Vector-Valued Functions
Term | Definition | ||
|---|---|---|---|
| Vector-Valued Function | A function that maps real numbers to vectors in a vector space. Each input value corresponds to a vector output, making it a multivariable function where the output is a vector rather than a scalar. | ||
| Component Function | Refers to the individual functions that make up a vector-valued function. Each component function describes one element of the vector output as a function of the input variables. | ||
| Arc Length Function | Measures the length of a curve described by a vector-valued function. It is often used to calculate the distance traveled along a path in a space. | ||
| Derivative of Vector-Valued Function | A vector that represents the rate of change of the function with respect to the input variables. It is calculated by taking the derivatives of each component function separately. | ||
Differentiation of Multivariable Functions
Term | Definition | ||
|---|---|---|---|
| Generalized Chain Rule | An extension of the chain rule from single-variable calculus to multivariable functions. It states how to compute the derivative of a composite function, where both the outer and inner functions may depend on several variables. | ||
| Total Differential | Represents the change in the function's value resulting from infinitesimal changes in each of its variables. It is expressed as a sum of the partial derivatives of the function with respect to each variable, multiplied by the corresponding changes in those variables. | ||
| Tangent Plane | A plane that touches a surface at a given point and has the same slope as the surface does at that point. It serves as a local approximation of the surface near that point | ||
| Saddle Point | A point where the surface curves up in one direction and curves down in another direction. It is a critical point where the surface is neither a maximum nor a minimum. | ||
| Partial Derivative | The rate of change of the function with respect to a variable, holding all other variables constant. | ||
| Gradient | A vector that points in the direction of the greatest rate of increase of the function at a given point. Its magnitude represents the rate of increase, and its direction points uphill. | ||
| Lagrange Multiplier | Used to find the extrema of a function subject to one or more constraints. | ||
| Constraint | A condition that limits the set of possible values for one or more variables in a mathematical problem. | ||
Multiple Integrals
Term | Definition | ||
|---|---|---|---|
| Double Integral | An integral of a function of two variables over a region in the plane. It represents the volume under the surface defined by the function and above the region in the plane | ||
| Fubini's Theorem | States that if a function is integrable over a rectangular region in the plane (or a rectangular box in space), then the value of the double (or triple) integral is the same regardless of the order in which the integrals are performed. | ||
| Iterated Integral | The process of evaluating a multiple integral by performing a sequence of one-dimensional integrals. For example, a double integral can be computed by integrating the integrand first with respect to one variable and then with respect to the other variable over the specified region. | ||
| Jacobian | A matrix of partial derivatives that describes the rate at which changes in one set of variables correspond to changes in another set of variables. | ||
| Triple Integral | An integral of a function of three variables over a region in three-dimensional space. It represents the volume under the surface defined by the function and above the region in space. | ||
Vector Calculus
Term | Definition | ||
|---|---|---|---|
| Closed Curve | A curve or a path in a space whose starting point is the same as its ending point. Mathematically, it is a continuous mapping of a closed interval into a space. | ||
| Curl | Measures the rotation or "curling" behavior of the field at a point. Mathematically, it is a vector operator that describes the tendency of a vector field to circulate around a point. | ||
| Divergence | Measures the rate at which the field spreads out or diverges from a point. | ||
| Flux | A measure of the flow of a vector field through a surface. It is calculated as the surface integral of the dot product of the vector field and the surface normal vector over the surface. | ||
| Fundamental Theorem for Line Integrals | Establishes a relationship between the line integral of a vector field and the scalar potential function associated with the field. It states that if a vector field is conservative, then the line integral of the field between two points depends only on the endpoints and can be computed by evaluating the potential function at those points. | ||
| Green's Theorem | Relates a line integral around a simple closed curve in the plane to a double integral over the region enclosed by the curve. It establishes a connection between line integrals and double integrals, providing a useful tool for calculating areas and evaluating line integrals. | ||
| Line Integral | An integral of a function of one or more variables along a curve or a path. | ||
| Vector Field | A mathematical construct that assigns a vector to each point in a given space. | ||
| Stokes' Theorem | Relates the circulation of a vector field around a curve to the flux of the curl of the vector field through a surface bounded by the curve. | ||
| Surface Area | Represents the sum of the areas of all the individual facets or surfaces that make up the object. | ||
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