Vector Calculus: Gradient of Scalar and Vector Fields, Lecture notes of Mathematics

Download Vector Calculus: Gradient of Scalar and Vector Fields and more Lecture notes Mathematics in PDF only on Docsity!

MAT3100: MULTIVARIABLE CALCULUS

MWALE David

Contents

• 1 FUNCTIONS OF SEVERAL VARIABLES
• 2 Further Differential Calculus
• 3 Surfaces
• 4 Multiple Integration
• 5 Vector Analysis
  • 5.1 Scalar and Vector fields
• Bibliography

Chapter 2

Further Differential Calculus

Chapter 3

Surfaces

Chapter 5

Vector Analysis

5.1 Scalar and Vector fields

Definition 5.1.1. If every point p(x; y; z) of a region R of space has associated with it a scalar quantity say φ(x; y; z); then φ(x; y; z) is a scalar function and a scalar field is said to exist in the region R: Temperature is an example of a scalar field.

Definition 5.1.2. Definition: Let M and N be functions of two variables x and y defined on a plane region R: Then the function F defined by

F(x; y) = M(x; y) i + N(x; y) j

is called a vector field over R:

In a similar fashion, if M; N and P are functions of three variables x; y and z defined on a solid region π in space, then the function

F(x; y; z) = M(x; y; z) i + N(x; y; z) j + P(x; y; z) k

is called a vector field over π: We have already met one example of a vector field, that of the gradient. If

z = f(x; y) is differentiable at a point (x 0 ; y 0 ); then the gradient of f at (x 0 ; y 0 ) denoted by ∇ f is given by

∇ f(x 0 ; y 0 ) = fx(x 0 ; y 0 ) i + fy (x 0 ; y 0 ) j :

We define the gradient of the function of three variables in a similar way.

Example 5.1.3. Let f(x; y) = sin xy^2 + ex^2 y^3 : Calculate ∇ f(1; 1): Solution From f(x; y) = sin xy^2 + ex^2 y^3 ; it is easy to show that fx(x; y) = y^2 cos xy^2 + 2xy^3 ex^2 y^3 and fy (x; y) = 2xy cos xy^2 + 3x^2 y^2 ex^2 y^3 : Consequently,

∇ f(x; y) = (y^2 cos xy^2 + 2xy^3 ex^2 y^3 ) i + (2xy cos xy^2 + 3x^2 y^2 ex^2 y^3 ) j ;

so that ∇ f(1; 1) = (cos 1 + 2e) i + (2 cos 1 + 3e) j :

Example 5.1.4. Find the gradient vector of f(x,y,z)=zxy^ at the point (2; 2 ; 2): Solution Homework.

Theorem 5.1.5. 1. If F(x; y) = M(x; y) i + N(x; y) j is a continuously differ- entiable vector field in an open disc D centered at (x; y); then in D; F is the gradient of some function f(x; y) if and only if

My (x; y) = Nx(x; y):

In this case we say that the vector field is exact.

2. Let F(x; y; z) = P(x; y; z) i + Q(x; y; z) j + R(x; y; z) k be a continuously differentiable function on an open region R in R^3 ; then F is the gradient of some function f(x; y; z) if and only if

Py (x; y; z) = Qx(x; y; z); Pz(x; y; z) = Rx(x; y; z); Qz(x; y; z) = Ry (x; y; z):

Bibliography

Prescribed Readings

1. Grossman S.T., (1996) Multivariable Calculus, Linear Algebra and Differential Equations, 3rd ed. HBL Publishers. ISBN: 0 155 64751 2 Recommended Readings
2. Stewart J., (2009) Calculus 6th ed. Brooks/Cole Cengage Learning, ISBN: 0 495 38362 8
3. Wylie R.C. and Barret L.C. (1995). Advanced Engineering Mathematics. McGraw-Hill. Company. ISBN: 0 070 72206 4
4. Kreyzig E. (2011). Advanced Engineering Mathematics. 10th Ed. John Willey and Sons. ISBN: 0 470 45836 5