<<vector calculus>> Notes

<<vector calculus>>
by Paul C, Matthews

P4

Since the quantity of |b|*cosθrepresents the component of the vector B in thedirection of the vector A, the scalar a * b Can be thought of as the magnitudeof a multiplied by the component of B in the direction of a

P7

The general form of the equation of a plane is:r * a = constant.

P11

| E1 E2 E3 |
A x b=| A1 A2 A3 |
| B1 B2 B3 |

V =ωx R

P24

The equation of a line is:r = a +λu

The second equation of a line is:r x u = b = A x u

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1.4 Scalar Triple Product ([A, B, C])

The dot and the cross can be interchanged:[a, B, C]≡a * b x c = a x B * C

The vectors a, B and C can be permuted cyclically:a * b x c = b * c x a = c * A x B

The scalar triple product can be written in the form of a determinant:

| A1 A2 A3 |
A * b x c=| B1 B2 B3 |
| C1 C2 C3 |

If any of the vectors was equal, the scalar triple product is zero.

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1.5 Vector Triple Product A x (b x C)

A x (b x c) = (A * c) *b-(A * b) *c

(A x b) x c =-(b * c) *a + (c * a) *b

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1.6 Scalar fields and vectors fields

A scalar or vector quantity is said are a field if it is a function of position.

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2.2.3 Conservative vector fields

A vector field F is said to being conservative if it has the property, the line integral of f around any closed curve C I S Zero:

An equivalent definition was that F-conservative if the line integral of Falong a curve only depends on the endpoints of The curve, not in the Pathtaken by the curve

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2.3.2

3.1.2 Taylor series in more than one variable

3.2 Gradient of a scalar field

The Symbol∇can is interpreted as a vector differential operator,where the term operator means that∇only have a meaning When it acts on some other quantity.

Theorem 3.1

Suppose that a vector field F are related to a scalar fieldφby F =∇φand∇exists everywhere in some region D. Then f are conservative within d.conversely, if F is conservative, then F can be written as the gradient of a scalar field, F =∇φ.

If a vector field F is conservative, the corresponding scalar Fieldφwhich obeys F =∇φis called the potential (potential energy) for F .

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3.3.2 Laplacian of a scalar field

3.3.2 Laplacian of a scalar field

4.3 The alternating Tensorεijk

5.1.1 Conservation of Mass for a fluid

6.1 Orthogonal curvilinear coordinates

P100

Suppose a transformation is carried out from a Cartesian coordinate system (x1, x2, x3) to another coordinate system (U1, U2, U3)

e1 = (∂X/∂U1)/h1, H1 = | ∂X/∂U1 |

E2 = (∂X/∂U2)/h2, H2 = | ∂X/∂U2 |

E3 = (∂X/∂U3)/h3, h3 = | ∂x/∂u3 |

DS = H1 * H2 * DU1 * DU2

DV = H1 * H2 * h3 * DU1 * DU2 * DU3

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The relevant content is elaborated in the Calculus Tutorial (volume III) (by Fihkingorts) using the Jacobi type:

Chapter 16

$4. Variable transformation in double integrals

603. Transformation of the plane area

604. Example 1) (example of polar coordinates)

605. Representation of area in curve coordinates

607. Geometric deduction

609. Variable transformation in double integrals

17 Chapter curved surface area, curved area Division

619. Example 2 (Introduction a,b,c)

The existence and calculation of surface area of 626

629 Cases 14) calculation of spherical polar coordinates

Chapter 18 triple points and multiple integrals

Variable Transformation in triple integrals

655. Spatial transformation and curve coordinates

656 Cases 1 Cylindrical coordinates, example 2 spherical coordinates

657 volume representation under curvilinear coordinates (to derive volume elements under surface coordinates)

659 Geometric deduction

Variable transformation in 6,613-heavy integrals

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Summary of Chapter 6

The System (U1, U2, U3) is orthogonal if EI * ej =δij.

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7. Cartesian tensors

7.1 Coordinate transformations

A matrix with the property, which is the inverse is equal to its transpose, is said to be orthogonal.

So far we had only considered a two-dimensional rotation of coordinates. Consider now a general three-dimensional rotation. For a position vector x = x1e1 + x2e2 + x3e3,

X ' = e ' i * x (projection of x on e ' i) = e ' I * (e1*x1 + e2*x2 + e3*x3) = e ' I * ei*xi

XI = Lji * x ' J ....... ........... (7.6)

7.2 Vectors and Scalars

A quantity is a tensor if all of the free suffices transforms according to the rule (7.4). Lij * Lkj =δik

7.3.3 Isotropic tensors

The Tensorsδij Andεijk has a special property. Their components is the same in all coordinate systems. A tensor with the said to be isotropic.

7.4 Physical examples of tensors

7.4.1 Ohm ' s Law

This is Whyδik was said to being an isotropic tensor:it represents the relationship between both vectors that was always para Llel, regardless of their direction.

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8 Applications of Vector calculus

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8.5 Fluid Mechanics

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Example 8.12

Choosing the x-axis to being parallel to the channel walls, the velocity u hasthe form u = (u, 0, 0). As the fluid is incompressible (all points are the same speed (along the x-axis), ∇u = 0, so∂u/∂x = 0.

<<vector calculus>> Notes