scalar

Examples of scalar in the following topics:

• Multiplying Vectors by a Scalar

  • While adding a scalar to a vector is impossible because of their different dimensions in space, it is possible to multiply a vector by a scalar.
  • A scalar, however, cannot be multiplied by a vector.
  • To multiply a vector by a scalar, simply multiply the similar components, that is, the vector's magnitude by the scalar's magnitude.
  • Multiplying vectors by scalars is very useful in physics.
  • For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.

• Unit Vectors and Multiplication by a Scalar

  • Multiplying a vector by a scalar is the same as multiplying its magnitude by a number.
  • In addition to adding vectors, vectors can also be multiplied by constants known as scalars.
  • Examples of scalars include an object's mass, height, or volume.
  • When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar .
  • (iii) Increasing the mass (scalar) increases the force (vector).

• Scalars vs. Vectors

  • Physical quantities can usually be placed into two categories, vectors and scalars.
  • In contrast, scalars require only the magnitude.
  • Scalars differ from vectors in that they do not have a direction.
  • Scalars are used primarily to represent physical quantities for which a direction does not make sense.
  • This video introduces the difference between scalars and vectors.

• Introduction to Scalars and Vectors

  • Given this information, is speed a scalar or a vector quantity?
  • Speed is a scalar quantity.
  • Distance is an example of a scalar quantity.
  • Scalars are never represented by arrows.
  • (A comparison of scalars vs. vectors is shown in . )

• Addition and Subtraction; Scalar Multiplication

  • Matrix addition, subtraction, and scalar multiplication are types of operations that can be applied to modify matrices.
  • There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication.
  • In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction.
  • Scalar multiplication has the following properties:
  • Practice adding and subtracting matrices, as well as multiplying matrices by scalar numbers

• Matrix and Vector Norms

  • For scalars, the obvious answer is the absolute value.
  • The absolute value of a scalar has the property that it is never negative and it is zero if and only if the scalar itself is zero.
  • A norm is a function from the space of vectors onto the scalars, denoted by $\| \cdot \|$ satisfying the following properties for any two vectors $v$ and $u$ and any scalar $\alpha$ :

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Definition: A vector field $\mathbf{v}$ is said to be conservative if there exists a scalar field $\varphi$ such that $\mathbf{v}=\nabla\varphi$.
  • When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf{v}$.
  • For any scalar field $\varphi$: $\nabla \times \nabla \varphi=0$.

• Line Integrals

  • The function to be integrated may be a scalar field or a vector field.
  • The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
  • More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve.
  • For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:
  • The line integral over a scalar field $f$ can be thought of as the area under the curve $C$ along a surface $z = f(x,y)$, described by the field.

• Surface Integrals of Vector Fields

  • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
  • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
  • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
  • Based on this reasoning, to find the flux, we need to take the dot product of $\mathbf{v}$ with the unit surface normal to $S$, at each point, which will give us a scalar field, and integrate the obtained field as above.
  • Explain relationship between surface integral of vector fields and surface integral of a scalar field

• Linear Vector Spaces

  • The definition of such a space actually requires two sets of objects: a set of vectors $V$ and a one of scalars $F$ .
  • For our purposes the scalars will always be either the real numbers $\mathbf{R}$ or the complex numbers $\mathbf{C}$ .
  • Addition and scalar multiplication are defined component-wise:
  • This implies the uniqueness of the zero element and also that $\alpha \cdot 0 = 0$ for all scalars $\alpha$ .
  • And the minus element is inherited from the scalars: $[-f](t) = -f(t)$ .