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Chapter 9 Vector Calculus

Page history last edited by chenyi 13 years, 3 months ago


Chapter 9

Vector differential calculus. Grad, Div, Curl


1 Vectors in 2-Space and 3-Space

A scalar is a quantity that is determined by its magnitude; A vector is a quantity that is determined by both its magnitude and its direction, thus it is an arrow of a directed line segment. Sometimes, we denote vectors by lowercase boldface letters, Formula, etc. In handwriting, you may use arrows, for instance Formula (in place of a). 

Note. A vector of length 1 is called a unit vector.


1.1 Equality of vectors

Two vectors Formula and Formula are equal, written Formula, if they have the same length and same direction. Hence a vector can be arbitrarily translated; that is, its initial point can be chosen arbitrarily.



1.2 Components of a vector

We choose an Formula Cartesian coordinate system in space. Let Formula be a given vector with initial point Formula and terminate point Formula. Then


are called the components of the vector Formula with respect to that coordinate system. and we simply write 


Note: The zero vector 0 has length Formula and no director.

Also if it is given Formula, that means Formula for the components.

Example 1.2. The vector Formula with the initial point Formula and terminal point Formula has the components


Hence Formula

Exercise 1.2. Find the component and magnitude of Formula with the initial point (-1,3) and terminal point (-4, 2).


1.3 Addition of Vectors

The sum Formula of two vectors Formula and Formula is obtained by adding the corresponding components.




1.3.1 Basic Properties of Vector Addition



1.4 Scalar Multiplication (Multipulication by a Number)

The product Formula of any vector Formula and any scalar Formula (real number Formula) is Formula.


1.4.1 Basic Properties of Scalar Multiplication


Example 1.4. Given Formula and Formula, find Formula



Exercise 1.4. Given Formula and Formula, find Formula, Formula, Formula.


1.5 Unit Vectors Formula, FormulaFormula

Besides Formula, another popular way of writing vector is


where Formula, Formula, Formula are the unit vectors in the positive directions of the axes of a Cartesian coordinate system.




2 Inner Product (Dot Product) of Vector

The inner product Formula of two vectors Formula and Formula is the product of their lengths times the cosine of their angle.

                                             Formula                         (1)

Note: Formula if Formula or Formula

The angle Formula, Formula, between Formula and Formula is measured when the initial points of the vectors coincide.

For Formula and Formula,

                                             Formula                         (2)



2.1 Orthogonality

A vector Formula is called orthogonal to a vector Formula if Formula. Then Formula is also orthogonal to Formula, and we call Formula and Formula orthogonal vectors.

The inner product of two nonzero vectors is 0 if and only if these vectors are perpendicular.


2.2 Length and Angle

From (1) with Formula gives Formula. Hence

                                                  Formula                         (3)

From (1) and (3) we obtain for the angle Formula between two nonzero vectors.

                                                  Formula                         (4)


Example 2.2. Find the inner product and the lengths of Formula and Formula as well as the angle between these vectors.



and (4) gives the angle 


Exercise 2.2. Let Formula and Formula . Find Formula, Formula and Formula.


2.3 Some properties of the vectors

From the definition we see that the inner product has the following properties,


Note. Gemetrically, (g) with Formula says that one side of a triangle must be shorter than the sum of the other two sides.


3 Application of the inner product


3.1 Work done by a force expressed as an Inner product

Let the body be given a displacement Formula. Then the work done by a constant force Formula in the displacement is defined as


If Formula and Formula are orthogonal, then the work done is zero.



3.2 Projection (Component) of a vector in a given direction

The projection of a vector Formula in a given direction of a vector Formula, is defined by

                                                  Formula                         (5)

where p is the length of the projection of Formula on the straight line l parallel to Formula.



Multiplying (5) by Formula, we have

                                                  Formula                         (6)

If Formula is a unit vector, then (6) simply gives

                                                  Formula                         (7)


The figure below shows the projection p of Formula in the direction of Formula and the projection Formula of Formula in the direction of Formula.



3.3 Orthogonal Straight Lines in the Plane

Example 3.3.



Find the straight line Formula through the point P: (1, 3) in the xy-plane and perpendicular to the straight line Formula.


First we express a general straight line


as Formula with Formula. By (2), now the line Formula through the origin and parallel to Formula

By orthogonality, the vector Formula is perpendicular to Formula. That means Formula is perpendicular to Formula and Formula because Formula and Formula are parallel. Formula is called the normal vector of Formula.

Now a normal vector of the given line Formula is Formula. Thus Formula is perpendicular to Formula if Formula, for instance, if Formula.

Hence Formula is given by Formula. It passes through P: (1,3) when Formula. Therefore, the required line is Formula.

Exercise 3.3. Find the straight line Formula through the point P: (-3, -6) in the xy-plane and perpendicular to the straight line Formula.


3.4 Normal Vector to a Plane

Example 3.4. Find a unit vector perpendicular to the plane Formula


By (2), we may write any plane in space as


where Formula.

The unit vector in the direction of Formula is



Dividing Formula by Formula, we have


From (7), we see that p is the projection of Formula in the direction of Formula. This projection has the same constant value Formula for the position vector Formula of any point in the plane. This holds if and only if Formula is perpendicular to the plane. Formula is called a unit normal vector of the plane (the other being Formula).

Furthermore, from this and the definition of projection it follows that |p| is the distance of the plane from the origin. Representation Formula is called Hesse's normal form of a plane. In our case, Formula, Formula, Formula, Formula, and the plane has the distance 7/6 from the origin.

Exercise 3.4. Find a unit vector perpendicular to the plane Formula.


4 Vector Product (Cross Product)

The vector product (also called cross product or outer product) Formula. (read Formula) of two vectorsFormula  is the vector


as follows. If Formula have the same or opposite direction, or if Formula then Formula. In any other case Formula has the length

                                                  Formula                         (8)

Formula is the angle between Formula. The direction of Formula is perpendicular to both Formula and such that Formula, in this order, form a right-hand triple.


In components, let Formula and Formula.Then Formula has the components



Formula                         (10)

Example 4.1. Vector Product

For the vector product Formula of Formula and Formula we obtain,




Note. Vector products of the standard basis vectors 

                                                  Formula                         (11)

4.1 General Properties of Vector ProductsFormula

Exercise 4.1. Try to prove (b), (c) and (d).


5 Typical Applications of vector products

5.1 Moment of a force

Example 5.1.

In mechanics the moment $m$ of a force Formula about a point Q is defined as the product Formula, where d is the distance between Q and the line of action L of Formula. If Formula is vector from Q to any point A on L, then Formula and


Since Formula is the angle between Formula, we see from (8) that Formula. The vector 


is called the moment vector or vector moment of Formula about Q. Its magnitude is m. If Formula, its direction is that of the axis of the rotation about Q that Formula has the tendency to produce. This axis is perpendicular to both Formula.


5.2 Velocity of a rotating body

Example 5.2.

A rotation of a rigid body B in space can be simply and uniquely described by a vector Formula as follows. The direction of Formula is that of the axis of rotation and such that the rotation appears clockwise if one looks from the initial point of Formula to its terminal point. The length of Formula is equal to the angular speed Formula of the rotation, that is, the linear (or tangential) speed of a point of $B$ divided by its distance from the axis of rotation.

Let $P$ be any point of B and d its distance from the axis. Then P has the speed Formula. Let Formula be the position vector of P referred to a coordinate system with origin 0 on the axis of rotation. Then Formula, where Formula is the angle between Formula. Therefore,


From this and the definition of vector product we see that the velocity vector Formula of P can be represented in the from


This simple formula is useful for determining Formula at any point B.


6 Scalar triple product

The most important product of vectors with more than two factors is the scalar triple product or mixed triple product of three vectors Formula, Formula, Formula. It is denoted by Formulaand defined by

                                                  Formula                         (12)

Note. Because of the dot product it is a scalar.


6.1 Properties and applications of scalar triple products



Exercise 6.1. Try to use Figure 13 to prove (b).


Example 6.1. Tetrahedron

A tetrahedron is determined by three edge vectors Formula, as indicated in Fig.14. Find its volume when Formula.


Solution. The volume V of the parallelepiped with these vectors as edge vectors is the absolute value of

the scalar triple product

Ex 6.1.1. Try to obtain the scalar triple product



Hence V=72. The minus sign indicates that if the coordinates are rght-handed, the triple Formula is left-handed. The volume of a tetrahedron is Formula of that of the parallelepiped[1], hence 12.


7 Scalar functions and vector functions


7.1 Scalar function

Scalar function f maps a point P=(x,y,z) in space to a scalar value


and its value is dependent on P.


We said f defines a ''scalar field" in the domain space.

Example 7.1. The distance f(P) of any point P(x,y,z) from a fixed point Formula in space is a scalar function whose domain of definition is the whole space.


f(P) defines a scalar field in space.


7.2 Vector function

Vector function Formula maps each point P=(x,y,z) to a vector, i.e. 


and its value is dependent on the point P in space.

We say that Formula defines a ''vector field" in the domain space.

Example 7.2. Let a particle A of mass M be mixed at a point Formula ans let a particle B of mass m be free to take up various positions P in space. Then A attracts B. 

Consider the gravitational force Formula between A and B,


when Formula and Formula.

This vector function describes the gravitational force action on B.


if we consider the gravitational potential energy,


which is a scalar function.


8 Vector calculus

8.1 Derivative of a vector function

A vector function Formula is said to be differentiable at a point t if the following limit exists:

                                                  Formula                         (13)

This vector Formula is called the derivative of Formula.


In components with respect to a given Cartesian coordinate system,

                                             Formula                         (14)

Hence Formula is obtained by differentiating each component separately.


Example 8.1. Find the derivative of Formula.




The familiar differentiation rules continue to hold for differentiating vector functions, for instance,


and in particular,



Note. Chain rule can also apply in differentiating vector functions. 

Let Formula and Formula, Formula, Formula, Then


The first partial derivatives with respect to u and v are


For example, if Formula and we define polar coordinates Formula, Formula by Formula, Formula, then the first derivatives with respect to Formula and Formula are 


Exercise 8.1. Find Formula when Formula.

The sides of a triangle are functions of time as Formula, Formula. Computer the rate of change of the area of the triangle at t=1.



8.2 Partial derivatives of a vector function

Suppose that the components of a vector function


are differentiable functions of n variables Formula. Then the partial derivative of Formula with respect toFormula  is denoted by Formula and is defined as the vector function



Example 8.2. Let Formula. Then


Exercise 8.2. Find the first partial derivatives of Formula.

The amount of water in a river


Find the velocity vector of the water.



9 Parametric Representation of curves

We usually represent a curve C in space in terms of its projections into the xy-plane and xz-plane, that is


Also it can be expressed in the following parametric form with parameter t:


One can also interpret (18) as a vector function Formula


To each value Formula there corresponds a point of C with position vector Formula with coordinates Formula.


Example 9.1 Circle

The unit circle Formula in the xy-plane can be represented parametrically by


where Formula. Indeed, Formula.


Example 9.2 Straight line

A straight line L with direction equal to that of a vector Formula, which passes through a point A with position vector Formula can be parametrically represented as


For instance, the straight line in the xy-plane through A:(3,2) having slope 1 is



Example 9.3 Circular Helix

A Helix C can be represented by the vector function 


It lies on the cylinder Formula. If c>0, the helix is shaped like a right-handed screw. If c<0, it looks a left-handed screw. If c=0, then it becomes a circle.



Exercise 9.1. Consider a function Formula, prove that Formula represents the surface of a unit sphere Formula.


10 Tangent to a curve

The tangent to a curve C at a point P of C is the limiting position of a straight line L that passes through P and a point Q of C, as Q approaches P along C.

If C is given by Formula, and P and Q correspond to Formula and Formula, then a vector in the direction of L is


In the limit this vector becomes the derivative


If Formula is a tangent vector of C at P and the corresponding unit vector is the unit tangent vector 


Then the tangent to C at P is given by

                                                  Formula                         (19)

where Formula is the position vector of P and w is the parameter in (19).



Example 10.1 Find the tangent to the ellipse Formula at Formula.

Solution. The parametric representation of the ellipse is


The derivative is Formula And $P$ corresponds to Formula since Formula.

From (19), the tangent should be



Exercise 10.1 An object is rotating around the point Formula at a distance r and angular velocity Formula. At time t=0, its location is (a+r,b).



11 Length of a curve


Let Formula, Formula, represent a curve C, its length is given by


If we replace the fixed b in (20) with a variable t, the integral becomes a function of t, denoted by Formula and called the arc length function or simply the arc length of C.

                                                  Formula                         (21)

Example 11.1 Consider the helix Formula, find the length of segment between t=0 and t=Formula.

Solution. By (20),


Exercise 11.1 A space craft file in a helix as Formula.



12 Gradient of a scalar field. Directional derivative


12.1 Gradient

The gradient of a given scalar function f(x,y,z) is denoted by grad f or Formula and is the vector function defined by

Formula                         (22)

Here x, y, z are Cartesian coordinates in a domain in 3-space in which f is defined and differentiable.


Example 12.1 Find the gradient of Formula.

Solution. Formula.

Exercise 12.1 What is Formula?


12.2 Directional derivative

The directional derivative Formula or Formula of a function f(x,y,z) at a point P in the direction of a vector Formula is defined by


Here Q is a variable point on the straight line L in the direction of Formula, and |s| is the distance between P and Q. Also, s>0 if Q lies in the direction of Formula, s<0 if Q lies in the direction of Formula, and s=0 if Q=P.


Consider Formula is a unit vector, i.e. Formula. Then L is given by

                                                  Formula                         (24) 

where Formula the position vector of P.

By (23) and apply the chain rule, we obtain

                                                  Formula                         (25)

By differentiating (24), it gives Formula. Hence (25) is simply the inner product of Formula and Formula; that is,

                                                  Formula                         (26)

Note. If the direction is given by a vector Formula of any length Formula, then



Example 12.2 Find the directional derivative of Formula at P:(2,1,3) in the direction of Formula.

Solution. Formula gives at P the vector Formula. Since Formula,


The minus sign indicates that at P the function f is decreasing in the direction of Formula.


Exercise 12.2 Find the directional derivative of $f=xyz$ at $P:(-1,1,3)$, $\vec{a}=[1,-2,2]$.


12.3 Gradient as the direction of maximum change

Since the directional derivative along direction of Formula is given by


 where Formula is the angle between Formula and Formula.

Observe that, for a given f, Formula is maximized when Formula, i.e. when Formula is parallel to Formula. Thus, at any point (x,y,z), the direction of the gradient, i.e. the direction of the vector Formula, is indeed the direction of maximum change of the function f at that point.



12.3.1 Computing Maxima/Minima via the Gradient


Since the gradient vector of a function points in the direction of maximum increase, hence the maximum or minimum value of an unconstrained function occurs when the gradient vector equals zero. 


Unconstrained optimization: The maximum/minimum of a function occurs when Formula=0.


Example 12.3.1: Consider the function Formula

Its gradient equals Formula.

Setting this vector to zero gives the equations FormulaFormula, and Formula. Solving these equations together gives us that (x,y,z) = (1,0,0) is a maximum/minimum of the function. To check whether in fact it is a maximum or minimum requires more work...


Constrained optimization: The maximum/minimum of a function occurs when Formula=0.

Suppose, further, one is given constraints on the variables of f, then one needs to check the boundary as well, as in the example below. 


Example 12.3.2: Consider the function in Example 12.3.1, find its maximum/minimum in the region Formula

The figure above shows the region. First, we need to check whether the solution to the unconstrained problem is feasible or not. As it can be seen, (1,0,0) is not in the region A, and hence it is not feasible.

Next, we need to consider all four edge line segments E1~E4.

E1)Formula, then Formula. Let the gradient equal zero, we get y=0. It is not in the line E1, so it is not feasible.

E2)Formula, then Formula. Let the gradient equal zero, x=1. Since (1,2,0) is on the line, it is a possible solution.

E3)Formula, then Formula. Let the gradient equal zero, y=0. It is not in the line E3, so it is not feasible.

E4)Formula, then Formula. Let the gradient equal zero, x=1. Since (1,1,0) is on the line, it is a possible solution.

Finally, we need to consider all four corner points P1, P2, P3 and P4. That is, (1,2,0), (1,1,0), (0,1,0), (0,2,0). After comparing all possible solutions, we find the minimum is (1,1,0).


Checking for maxima or minima: Just as in the scalar case, the double derivative of the function tells us whether the function attains a maximum or a minimum at a point, similarly for functions of many variables, a similar concept called the Hessian helps.


(Advanced) Hessian: is the square matrix of second-order partial derivatives of a function. Consider the function  Formula. Its Hessian is



Note that, if the second derivatives of f are all continuous in a neighborhood D, then the Hessian of is a symmetric matrix throughout D. Let the gradients equal zero, we get x=1, y=0. If the determinant of Hessian at (1,0) is positive, then f attains a local minimum at (1,0). If the determinant is negative, then (1,0) is a local maximum. If the determinant equals zero, then (1,0) is a saddle point. In this case, the determinant is 24, so it is a local minimum.  


For higher dimensional Hessian, if the Hessian is positive definite at point A , then f attains a local minimum at A. If the Hessian is negative definite at A, then f attains a local maximum at A. If the Hessian has both positive and negative eigenvalues, then A is a saddle point  for f.

Positive definite: An n × n real symmetric matrix M is positive definite if  Formula   for all non-zero vectors z with real entries (z \in \mathbb{R}^n), where zTdenotes the transpose of z.


Try to find out Example 12.3.1, is it a maximum or minimum?





12.4 Gradient as surface normal vector

Let f be a differentiable scalar function in space and f(x,y,z)=c=const represent a surface S. Now let C be a curve on S through a point P of S. Then C has a representation Formula. For C to lie on the surface S, the components of Formula must satisfy f(x,y,z)=c that is,

                                                  Formula                         (27)


Recall a tangent vector of C is


And the tangent vectors of all curves on S passing through P will generally form a plane, called the tangent plane of S at P. The normal of this plane is called the surface normal to S at P. A vector in the direction of the surface normal is called a surface normal vector of S at P. We can obtain such a vector by differentiating (27) w.r.t t,


Hence if Formula at P of S is not the zero vector, then Formula is perpendicular to all the vector Formula in the tangent plane, so that it is a normal vector of S at P.



Example 12.4 Gradient as surface normal vector. Cone

Find a unit normal vector Formula of the cone of revolution Formula at the point P:(1,0,2).

Solution. We have Formula.

Ex 12.4.1 Find Formula and Formula.




Ex 12.4.2 Find Formula.




Formula points downward since it has a negative $z$-component. The other unit normal vector the cone at P is Formula.


Exercise 12.3. Find a normal vector of the surface Formula, P:(4,3,8).


13 Divergence of a vector field

Let Formula be a differentiable vector function, where x, y, z are Cartesian coordinates, and let Formula be the components of Formula. Then the function

                                                  Formula                         (28)


is called the divergence of Formula or the divergence of the vector field defined by Formula.


Example 13.1 Find the div Formula when Formula.

Solution. div Formula.

Another common motion for the divergence is


Note. The divergence div Formula is a scalar function, i.e. its values depend only on the points in space and onFormula but not on the choice of the coordinates in (28).

The physical meaning of the Divergence is a important relation

                                                  Formula                         (29) 

(Formula: a velocity vector of a the motion of a fluid in some region.)

which is called the condition for the conservation of mass or the continuity equation of a compressible fluid flow.


If the flow is steady, that is, independent of time, then Formula and the continuity equation is

                                                  Formula                         (30)

If the density Formula is constant, so that the fluid is incompressibility, then equation (30) becomes

                                                  Formula                         (31)

This relation is known as the condition of incompressibility. It expresses the fact that the balance of outflow and inflow for a given volume element is zero at any time. Roughly speaking, the divergence measures "outflow minus inflow" for some volume of object.


Exercise 13.1 If the velocity vector of water in a river is Formula, write u as a function of x,y,z.


14 Curl of a vector field

Let Formula be a differentiable vector function of the Cartesian coordinates x,y,z. Then the curl of the vector function Formula or of the vector field given by Formula is

                              Formula                         (32)

This is the formula when x,y,z are right-handed. If they are left-handed, the determinant has a minus sign in front.

The curl also has an important physical meaning and it is related to the rotation of an object or fluid.

Note. Curl Formula is a vector. That is, it has a length and direction that are independent of the particular choice of a Cartesian coordinate system in space.

Example 14.1 Let Formula with right-handed x,y,z. Then (32) gives 


Exercise 14.1 Let rotation vector Formula of a rotating body be (0,0,Formula),



15 Laplacian of a scalar field

Let f(x,y,z) be a scalar function, the laplacian of f denoted by Formula is defined as

                                      Formula                         (33)

Note. One application is the Laplace's equation 


It is the most important partial differential equation in physics.

Example 15.1 Find Formula where Formula.

Solution. Formula.

Exercise 15.1 Prove that gravitational/electric potentials, both of which have the form Formula satisfy Formula. (Hint: Formula and use chain rule.)


16 Some basic formulas for grad, div, curl

Let Formula be twice differentiable scalar functions and Formula be twice differentiable vector functions.


Exercise 16.1 Prove (19) - (28).


  1. http://en.wikipedia.org/wiki/Tetrahedron\#Volume

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