1 Vector and Tensor Calculus

Vector and Tensor Calculus BME 35200

Basic Concepts

  • Definition of a Vector:

    • A vector is defined as a mathematical entity that has both a magnitude (length) and a direction.

    • Represented as π‘Žβƒ‘.

    • The length of the vector is denoted as |π‘Žβƒ‘| and equals the length of the arrow.

    • Length is positive (|π‘Ž|) or negative (-|π‘Ž|) depending on its sign.

    • The unit vector 𝑒⃑ has a length of 1.

    • The zero vector (π‘Žβƒ‘ = 0) has a length of zero.

Vector Operations

  • Multiplication by a Scalar:

    • Multiplying vector π‘Žβƒ‘ by a positive scalar 𝛼 results in a new vector 𝑏⃑ that retains the original direction but has a different magnitude.

  • Vector Addition:

    • The sum of two vectors π‘Žβƒ‘ and 𝑏⃑ produces a new vector 𝑐⃑ equal to the diagonal of the parallelogram they span.

Inner Product

  • Dot Product:

    • The inner product (dot product) of two vectors yields a scalar quantity.

    • Properties:

      • Commutative: π‘Žβƒ‘ β€’ 𝑏⃑ = 𝑏⃑ β€’ π‘Žβƒ‘

    • It defines the length of a vector; if a vector is perpendicular, their inner product is zero.

Cross Product

  • Vector Product:

    • Cross product of vectors π‘Žβƒ‘ and 𝑏⃑ produces a new vector 𝑐⃑, which is perpendicular to both π‘Žβƒ‘ and 𝑏⃑.

    • Vectors π‘Žβƒ‘, 𝑏⃑, and 𝑐⃑ form a right-handed system.

    • The magnitude of the vector product equals the area of the parallelogram formed by π‘Žβƒ‘ and 𝑏⃑.

Right-Handed System

  • Definition:

    • If a corkscrew rotates from vector π‘Žβƒ‘ to vector 𝑏⃑, it moves in the direction of vector 𝑐⃑, creating a right-handed system.

Triple Product

  • The vector product of a vector with itself results in the zero vector.

  • The vector product is not commutative; the order matters.

    • Example: 𝑏⃑ Γ— π‘Žβƒ‘ = - (π‘Žβƒ‘ Γ— 𝑏⃑).

  • Triple Product: The scalar volume of the parallelepiped spanned by three vectors.

Tensor Product

  • Dyadic Product:

    • The tensor product of vectors π‘Žβƒ‘ and 𝑏⃑ defines a linear transformation operator (dyad) denoted as π‘Žβƒ‘π‘.

    • When applied to the vector 𝑝⃑, it transforms it into another vector along the direction of π‘Žβƒ‘.

Basis in Three-Dimensional Space

  • A set of three vectors is a basis if their triple product is non-zeroβ€”indicating they are non-coplanar.

  • Orthogonal Basis: Achieved if the basis vectors are mutually perpendicular.

  • Orthonormal Basis: Achieved if the orthogonal basis vectors have unit length.

  • A Cartesian basis is an orthonormal right-handed basis, location-independent in three-dimensional space.

Vector Decomposition

  • Any vector can be decomposed into components along three basis vectors (e.g., 𝑒⃑%, 𝑒⃑&, 𝑒⃑').

    • Represented as a sum: π‘Žβƒ‘ = π‘Žβ‚π‘’βƒ‘% + π‘Žβ‚‚π‘’βƒ‘& + π‘Žβ‚ƒπ‘’βƒ‘'.

    • Components can be organized in a column matrix notation.

Basis Representation Variances

  • Different bases yield different column representations for the same vector:

    • π‘Žβƒ‘'s representation can change based on the chosen basis.

Vector Operations in Column Form

  • Common operations in vector calculus can be expressed as column (matrix) operations:

    • Scalar multiplication, vector addition, inner product, vector product, dyadic product.

Examples

  • Example 1: Calculate inner and vector products for vectors π‘Žβƒ‘ = 𝑒⃑% + 2𝑒⃑& and 𝑏⃑ = 2𝑒⃑% + 5𝑒⃑&.

  • Example 2: Test if vectors πœ€βƒ‘