Syntax
diff(f, var, n)
computes
the nth derivative
of f with respect
to var.
Example1
$$\text{Find all the first order and second order partial derivatives of the function }u=e^{\frac{x}{y}}.$$ $$\text{Also, find }\frac{\partial u}{\partial x}\text{ at }x=2, \frac{\partial u}{\partial x}\text{ at }y=2 \text{ and } \frac{\partial u}{\partial y}\text{ at }x=2, y=3.$$
clear
clc
syms x y
%Define U
U(x,y)=exp(x/y);
%First derivatives
Ux=diff(U,x);
Uy=diff(U,y);
%Second derivatives
Uxx=simplify(diff(Ux,x));
Uyy=simplify(diff(Uy,y))
Uxy=simplify(diff(Ux,y))
Uyx=simplify(diff(Uy,x))
U1=Ux(2,y);
U2=Ux(x,2);
U3=Uy(2,3);
fprintf('\n
Ux(2,y)=%s; \t Ux(x,2)=%s; \t Uy(2,3)=%f \n',U1,U2,U3)
Output
The first order partial
derivatives are
Ux = exp(x/y)/y
Uy = -(x*exp(x/y))/y^2
The second order partial
derivatives are
Uxx = exp(x/y)/y^2
Uyy = (x*exp(x/y)*(x +
2*y))/y^4
Uxy = -(exp(x/y)*(x +
y))/y^3
Uyx = -(exp(x/y)*(x + y))/y^3
Ux(2,y)=exp(2/y)/y; Ux(x,2)=exp(x/2)/2; Uy(2,3)=-0.432830
Example 2
$$\text{If }z=\frac{1}{\sqrt(y^2-2xy+1)}\text{ then prove that }x\frac{\partial z}{\partial x}-y\frac{\partial z}{\partial y}=y^2z^3.$$
clear all
clc
syms z x y
z=(1-2*x*y+y^2)^(-1/2);
zx=diff(z,x);
zy=diff(z,y);
LHS=simplify(x*zx-y*zy)
RHS=simplify(y^2*z^3)
if(LHS==RHS)
fprintf('\n The given condition is
satisfied\n');
else
fprintf('\n The given condition is not
satisfied')
end
Output
LHS =
y^2/(- 2*x*y + y^2 +
1)^(3/2)
RHS =
y^2/(- 2*x*y + y^2 +
1)^(3/2)
The given condition is satisfied
Jacobian
jacobian([u1,u2,…],[x1,x2,…])
computes the
Jacobian of a vector function which is a matrix of the partial derivatives of
that function
·
To find the determinant find
the determinant of the resultant Jacobian matrix.
Example 1
$$\text{If }u=x^2-2y\text{ and }v=x+y\text{ then find the Jacobian }\frac{\partial (u,v)}{\partial (x, y)}.\text{ Also, find the value of}$$ $$\text{Jacobian at }(1, 0).$$
clear all
clc
syms x y u
v
u(x,y)=input('Enter
u(x,y):');
v(x,y)=input('Enter
v(x,y):');
J(x,y)=simplify(det(jacobian([u,v],[x,y])));
fprintf('\n
J((u,v)/(x,y))= %s \n', J);
p=input('\n
Enter the point in the form [a,b] : ');
val=J(p(1),p(2));
fprintf('\n
J((u,v)/(x,y)) at (%d,%d) is %d \n', p(1),p(2), val)
Output
Enter u(x,y):x^2-2*y
Enter v(x,y):x+y
J((u,v)/(x,y))= 2*x + 2
Enter the point in the form
[a,b] : [1,0]
J((u,v)/(x,y)) at (1,0) is 4
Example 2
$$\text{If }u=x^2-y^2\text{ and }v=2xy\text{ where }x=r \cos(t)\text{ and } y=r \sin(t)\text{ then find the Jacobian }\frac{\partial(u,v)}{\partial(x,y).}$$ $$\text{Also, show that }\frac{\partial(u,v)}{\partial(r,t)}=4r^3\text{ and hence find the value of Jacobian }\frac{\partial(u,v)}{\partial(r,t)}\text{ at }(3,1).$$
clear all
clc
syms x y u
v r t
u(x,y)=input('Enter
u(x,y):');
v(x,y)=input('Enter
v(x,y):');
J1(x,y)=det(jacobian([u(x,y),v(x,y)],[x,y]));
fprintf('\n
J((u,v)/(x,y))= %s \n', J1);
x=input('\n
Enter x(r,t):');
y=input('Enter
y(r,t):');
J2(r,t)=simplify(det(jacobian([u(x,y),v(x,y)],[r,t])));
fprintf('\n
J((u,v)/(r,t))=%s \n', J2)
p=input('\n
Enter the point in the form [a,b] : ');
val=J2(p(1),p(2));
fprintf('\n
J((u,v)/(r,t)) at (%d,%d) is %d \n', p(1),p(2), val)
Output
Enter u(x,y):x^2-y^2
Enter v(x,y):2*x*y
J((u,v)/(x,y))= 4*x^2 +
4*y^2
Enter x(r,t):r*cos(t)
Enter y(r,t):r*sin(t)
J((u,v)/(r,t))=4*r^3
Enter the point in the form
[a,b] : [3,1]
J((u,v)/(r,t)) at (3,1) is
108
Example 3
$$\text{If }u=x\sin(y)\cos(z), v=x\sin(y)\sin(z)\text{ and } w=x\cos(y)\text{ then find }\frac{\partial(u,v,w)}{\partial(x,y,z)}$$
clear all
clc
syms x y z
u v w
u(x,y,z)=input('Enter
u(x,y,z):');
v(x,y,z)=input('Enter
v(x,y,z):');
w(x,y,z)=input('Enter
w(x,y,z):');
J(x,y,z)=simplify(det(jacobian([u,v,w],[x,y,z])));
fprintf('\n
J((u,v,w)/(x,y,z))= %s \n', J);
Output
Enter
u(x,y,z):x*sin(y)*cos(z)
Enter
v(x,y,z):x*sin(y)*sin(z)
Enter w(x,y,z):x*cos(y)
J((u,v,w)/(x,y,z))=
x^2*sin(y)
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