Differential Vector Calculus, Exams of Vector Analysis

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Differential Vector Calculus

Steve Rotenberg CSE291: Physics Simulation UCSD Spring 2019

Fields

• A field is a function of position x and

may vary over time t

• A scalar field such as s ( x , t ) assigns a

scalar value to every point in space.

An example of a scalar field would be

the temperature throughout a room

• A vector field such as v ( x , t ) assigns a

vector to every point in space. An

example of a vector field would be the

velocity of the air

Gradient

• The gradient is a generalization of the concept of a derivative 𝛻𝑠 = 𝜕𝑠 𝜕𝑥 𝜕𝑠 𝜕𝑦 𝜕𝑠 𝜕𝑧 𝑇
• When applied to a scalar field, the result is a vector pointing in the direction the field is increasing and the magnitude indicates the rate of increase
• In 1D, this reduces to the standard derivative (slope)

Gradient

• The gradient 𝛻𝑠 is a vector that points “uphill” in the

direction that scalar field s is increasing

• The magnitude of 𝛻𝑠 is equal to the rate that s is

increasing per unit of distance

Divergence

• The divergence is positive where the field is expanding: 𝛻 ∙ 𝐯 > 0
• The divergence is negative where the field is contracting: 𝛻 ∙ 𝐯 < 0
• A constant field has zero divergence, as can many others: 𝛻 ∙ 𝐯 = 0

Curl

• The curl operator produces a new vector field that measures the rotation of the original vector field 𝛻 × 𝐯 =

𝑧 𝜕𝑦

𝑦 𝜕𝑧

𝑥 𝜕𝑧

𝑧 𝜕𝑥

𝑦 𝜕𝑥

𝑥 𝜕𝑦 𝑇

• For example, if the air is circulating in a particular region, then the curl in that region will represent the axis of rotation
• The magnitude of the curl is twice the angular velocity of the vector field

Laplacian

• The Laplacian operator is one type of second derivative of a scalar or vector field 𝛻 2 = 𝛻 ∙ 𝛻 = 𝜕 2 𝜕𝑥^2

𝜕 2 𝜕𝑦^2

𝜕 2 𝜕𝑧^2

• Just as in 1D where the second derivative relates to the curvature of a function, the Laplacian relates to the curvature of a field
• The Laplacian of a scalar field is another scalar field: 𝛻 2 𝑠 = 𝜕 2 𝑠 𝜕𝑥 2 + 𝜕 2 𝑠 𝜕𝑦 2

𝜕 2 𝑠 𝜕𝑧 2

• And the Laplacian of a vector field is another vector field 𝛻 2 𝐯 = 𝜕 2 𝐯 𝜕𝑥 2 + 𝜕 2 𝐯 𝜕𝑦 2 + 𝜕 2 𝐯 𝜕𝑧 2

Laplacian

• The Laplacian is positive in an area of the field that is surrounded by higher values
• The Laplacian is negative where the field is surrounded by lower values
• The Laplacian is zero where the field is either flat, linear sloped, or the positive and negative curvatures cancel out (saddle points)

Numerical Representation of

Fields

Computational Vector Calculus

• Now that we’ve seen the basic operations of

differential vector calculus, we turn to the

issue of computer implementation

• The Del operations are defined in terms of

general fields

• We must address the issue of how we

represent fields on the computer and how we

perform calculus operations on them

Uniform Grids

• Uniform grids are easy to deal with and tend to be computationally efficient due to their simplicity
• It is very straightforward to compute derivatives on uniform grids
• However, they require large amounts of memory to represent large domains
• They don’t adapt well to varying levels of detail, as they represent the field to an even level of detail everywhere

Uniform Grids

Hierarchical Grids

Hierarchical Grids