Cross Product
A vector operation on two vectors that produces a third vector perpendicular to both, with magnitude equal to the product of their magnitudes times the sine of the angle between them.
Dot Product
A scalar operation on two vectors that produces a number equal to the product of their magnitudes times the cosine of the angle between them. Used to find projections and determine orthogonality.
Eigenvalue
A scalar associated with a square matrix such that there exists a nonzero vector (eigenvector) where the matrix times that vector equals the scalar times that vector. Found by solving the characteristic equation.
Determinant
A scalar value computed from a square matrix that indicates whether the matrix is invertible (nonzero determinant) and relates to the scaling factor of the linear transformation.
Newton Laws of Motion
Three fundamental laws of classical mechanics: (1) an object at rest stays at rest unless acted on by a force, (2) F equals ma, (3) every action has an equal and opposite reaction.
Free Body Diagram
A sketch of a body isolated from its surroundings showing all external forces and moments acting on it. The essential first step in solving any statics or dynamics problem.
Equilibrium
A state where the sum of all forces and the sum of all moments acting on a body are zero. The basis of static analysis and a necessary condition for structures at rest.
Derivative
The instantaneous rate of change of a function with respect to its variable. Geometrically, the slope of the tangent line to the function curve at a given point.
Integral
The accumulation of a quantity over an interval. A definite integral gives the net area under a curve, while an indefinite integral gives the antiderivative of a function.
Differential Equation
An equation involving a function and its derivatives. Ordinary differential equations (ODEs) involve one independent variable; partial differential equations (PDEs) involve multiple independent variables.
Matrix Inverse
A square matrix that, when multiplied by the original matrix, produces the identity matrix. Exists only when the determinant is nonzero. Used to solve systems of linear equations.
Yield Strength
The stress at which a material begins to deform permanently (plastically). Below this stress, deformation is elastic and the material returns to its original shape when the load is removed.
Ultimate Tensile Strength
The maximum stress a material can withstand before failure in a tension test. Corresponds to the highest point on the engineering stress-strain curve.
Poisson Ratio
The ratio of lateral strain to axial strain when a material is loaded in one direction. A dimensionless material property, typically between 0.25 and 0.35 for metals.
Conservation of Energy
A fundamental principle stating that energy cannot be created or destroyed, only converted from one form to another. The basis for energy balance equations in all engineering disciplines.
Dimensional Analysis
A technique for checking equations and deriving relationships by ensuring dimensional consistency. The Buckingham Pi theorem uses it to reduce the number of variables in a problem.
Significant Figures
The number of meaningful digits in a measured or calculated value. Rules for arithmetic operations preserve the precision of the least precise measurement in the calculation.
Newton-Raphson Method
An iterative numerical method for finding roots of equations. Uses the tangent line at an initial guess to converge on the root. Requires the function and its derivative.