Vector Calculus Quiz

Vector calculus is a type of advanced mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space $R^3$.

Vector calculus has practical applications in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.

Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, with most of the notation and terminology established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis.

The broader subject that includes vector calculus is multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.

A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity.

Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. The study of differential equations consists mainly of the study of their solutions, and of the properties of their solutions.

Differential equations came into existence with the invention of calculus by Newton and Leibniz. The derivatives in differential equations represent their rates of change, which are fundamental concepts in calculus.

Only the simplest differential equations are soluble by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

The solutions of differential equations may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.