Calculo 2 De dos variables_9na Edición - Ron Larson

1058 CAPÍTULO 15 Análisis vectorial
1058 Chapter 15 15 Vector Analysis
15.1 Campos vectoriales
15.1 Vector Fields
■ Comprender el concepto de un campo vectorial.
■ Determinar Understand the si
the un concept campo of of
vectorial of a vector es field.
conservativo.
■ Calcular Determine el whether
rotacional a vector
de un field field
campo is is is vectorial.
conservative.
■ Calcular Find Find the the curl la
curl divergencia of of of a vector de field.
un campo vectorial.
■ Find Find the the divergence of of of a vector field.
Campos vectoriales
Vector Fields
En el capítulo 12 se estudiaron funciones vectoriales que asignan un vector a un número
In real. In Chapter Se comprobó 12, 12, you you que studied las funciones vector-valued vectoriales functions—functions de números reales that that assign son útiles a a vector para torepre-
asentar a real real curvas number. y There movimientos you you saw saw a that that lo largo vector-valued de una curva. functions En este of of real capítulo real numbers se estudiarán are are useful
otros
in dos in representing tipos de funciones curves and and vectoriales motion along que asignan a a curve. un In In vector this this chapter, a un punto you you will en will el study plano two two
oaun
punto other types en el of of espacio. vector-valued Tales funciones functions—functions se llaman campos that that assign vectoriales a a to to (campos a a point in de in the the
vectores),
plane y or or son a a point útiles in in para space. representar Such functions varios tipos are are called de campos vector de fields, fuerza and and y campos they are are de useful
velocidades.
in representing various types of of force fields and and velocity fields.
in DEFINICIÓN DEFINITION DE OF OF UN VECTOR CAMPO VECTORIAL
FIELD
Un A vector campo field vectorial over sobre a plane una region región R
is is plana is a a function R es una F
that that función assigns F que a a vector
asigna un
vector Fx, Fx, y y
F(x, to to each y) a cada point punto in in
R. R.
en R.
Un A vector campo field vectorial over sobre a solid una region región Q
in in sólida space Qis is is en a a el function espacio F
that es that una assigns función aa
F que
asigna vector un Fx, Fx, vector y, y, y, z z zto F(x, to each y, z) point a cada in in Q. punto Q.
en Q.
NOTE NOTA NOTE
Aunque Although un a a vector campo field field vectorial consists está of of of constituido infinitely many por infinitos vectors, you vectores, you can can get get se a a puede good good idea obtener idea of of ofuna
what idea what aproximada the the vector field field de su looks looks estructura like like by by dibujando sketching varios several vectores representative representativos vectors
Fx, F(x, Fx, y y), y y
whose cuyos puntos initial
iniciales
points are son are
x, (x, x, y. y). y.
■
■
The El The gradiente es is is is one un one ejemplo example de of of un a a campo vector field. vectorial. For For example, Por ejemplo, if if if si
fx, fx, y y xx 2 2 y y 3xy 3xy 33
entonces then the the gradient el gradiente of of
fffde
f
fx, y y f x fx, f xx x, yi yi f y fx, f yy x, yj yj
2xy 3y 3y 33 i i i x x 22 9xy 9xy 22 j
j j
Campo Vector Vector vectorial field field in in in en the the el plane plane plano.
is es is is a a un vector campo field vectorial in in the the plane. en el plano. From Chapter Del capítulo 13, 13, the the 13, graphical la interpretación interpretation gráfica of of this this de este field
campo
is is a a es family una familia of of vectors, de vectores each of of cada which uno points de los in in cuales the the direction apunta en of of la maximum dirección increase
de máximo
crecimiento along the the surface a lo largo given de by by
la zzsuperficie fx, fx, y. y.
dada por z fx, y.
De Similarly, manera if if if similar, si
is
fx, fx, fx, y, y, y, z z z z x x 2 2 y y 2 2 z z 2
2
entonces then the the gradient el gradiente of of
fffde
f
fx, fx, y, y, y, z z z z f x f x, f x
x, x x, y, y, y, zi zi zi f y f x, f y
x, y x, y, y, y, zj zj zj f z f x, f z
x, z x, y, y, y, zk zk zk
2xi 2xi 2yj 2yj 2zk
2zk
Campo Vector Vector vectorial field field en in in in space el space
espacio.
is es is is a un a vector campo field vectorial in in space. en el Note espacio. that that the the Notar component que las funciones functions componentes for for this this particular para este vector
campo
vectorial field are are
2x, 2x, particular 2y, 2y,
and and
son 2z. 2z.
2x, 2y y 2z.
Un A vector campo field
vectorial
Fx, Fx, y, y, y, z z z Mx, y, y, y, zi zi Nx, y, y, y, zj zj Px, Px, y, y, y, zk zk
is es is is continuous en at at at un a a punto point if if si if and and y sólo only si if if cada if each una of of its de its sus component funciones functions componentes M, M,
N, N,
and and M, NP
is y is isP es
continua continuous en at at ese at that that punto.
point.