Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
Syllabus Vector Differentiation - Velocity and Acceleration - Gradient ...
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Engineering Mathematics - II - <strong>Vector</strong> Calculus - 2007<br />
Summary<br />
<strong>Vector</strong> function: A vector function of a scalar variable t is of the form<br />
<br />
f1(t)<br />
i f2<br />
(t) j<br />
f3(t)<br />
k<br />
<strong>Differentiation</strong> of a <strong>Vector</strong>:<br />
<br />
d f df <br />
<br />
1 df df<br />
i <br />
2<br />
j<br />
3<br />
k<br />
dt dt dt dt<br />
Equation of a space curve: Equation of a space curve is represented by<br />
<br />
r (t)<br />
<br />
x(t) i y(t) j z (t) k<br />
<br />
d r<br />
<strong>Velocity</strong> :<br />
dt<br />
space curve.<br />
is<br />
the<br />
<br />
velocity v<br />
<strong>and</strong> is<br />
along<br />
the<br />
tan gent<br />
to<br />
the<br />
Unit<br />
tan gent vector<br />
<br />
: t<br />
<br />
<br />
d r<br />
dt<br />
<br />
d r<br />
| |<br />
dt<br />
<br />
Unit normal : n <br />
|<br />
<br />
d t<br />
ds<br />
<br />
d t<br />
|<br />
ds<br />
is called the unit normal to the curve.<br />
Scalar field: A scalar corresponding to each (x,y,z) in a region R.<br />
<strong>Vector</strong> field: A vector corresponding to each (x,y,z) in a region R<br />
<br />
<br />
<br />
<br />
r f1<br />
(x, y, z) i f2<br />
(x, y, z) j f3(x, y, z) k<br />
<strong>Gradient</strong>: If<br />
<br />
is<br />
a scalar field<br />
then<br />
<br />
<br />
<br />
i<br />
x<br />
<br />
j<br />
y<br />
<br />
k<br />
z<br />
is<br />
called<br />
the<br />
gradient of .<br />
Unit Normal to the surface : If = c is a surface then<br />
<br />
| |<br />
<br />
<br />
n<br />
is<br />
the unit<br />
normal<br />
to the surface.<br />
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