Vector Calculus

Cartesian Coordinates

● Vector:

Cylindrical Coordinates

● Vector:

Spherical Coordinates

● Vector:

Differential Length, Area, and Volume

Cartesian

Cylindrical

Cylindrical

Spherical

Spherical

Line, Surface, and Volume Integrals

Drills

Ans

Drill

● Determine the total surface area enclosing a cylinder.

● Determine the total surface area enclosing a sphere.

Ordinary Derivative of Vectors

Let R(u) be a vector depending on a scalar variable u.

Ordinary Derivative of Vectors

Let R(u) be a vector depending on a scalar variable u.

Differentiation Formulas

Differentiation Formulas

Applications

Streamlines and Sketches of Fields

Streamlines (flux lines or direction lines) are short line segments drawn with arrows pointing in the direction of E and the spacing of lines is inversely proportional to the strength of the field.

Streamlines and Sketches of Fields

Illustration: Like charges

Illustration: Opposite Charges

Equation of Streamlines

Example

Consider the vector field

Find the equation of one particular streamline passing through P(-2,7,10).

Example

Ans. y = -3.5x

Drill 2.7

Find the equation of the streamline that passes through the point P(1,4,-2) in the field E=: (a) -8x/y ax + 4x2/y2 ay;

Drill 2.7

Find the equation of the streamline that passes through the point P(1,4,-2) in the field E=: (a) -8x/y ax + 4x2/y2 ay;

Ans. x2 + 2y2 = 33 ;

END