Partial Differentiation: Part 3

Advanced Numerical Problems on Partial Differentiation

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Problem 1: Laplace’s Equation

Problem: Verify that \( u = x^3 - 3xy^2 \) satisfies Laplace’s equation.

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\[ u_x = 3x^2 - 3y^2, \quad u_y = -6xy \] \[ u_{xx} = 6x, \quad u_{yy} = -6x \] \[ u_{xx} + u_{yy} = 6x - 6x = 0 \]

Hence verified.

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Problem 2: Total Differential

Problem: If \( z = e^{xy}\sin(x+y) \), find total differential \( dz \).

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\[ dz = z_x dx + z_y dy \] \[ z_x = e^{xy}[y\sin(x+y)+\cos(x+y)] \] \[ z_y = e^{xy}[x\sin(x+y)+\cos(x+y)] \] \[ dz = e^{xy}[y\sin(x+y)+\cos(x+y)]dx \] \[ + e^{xy}[x\sin(x+y)+\cos(x+y)]dy \]

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Problem 3: Directional Derivative

Problem: Find directional derivative of \( \phi=x^2y+yz^3 \) at (1,-1,1) in direction \(2\mathbf{i}-\mathbf{j}+2\mathbf{k}\).

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\[ \nabla \phi = (2xy)\mathbf{i} + (x^2+z^3)\mathbf{j} + (3yz^2)\mathbf{k} \] At (1,-1,1): \[ \nabla \phi = (-2)\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \] Direction vector magnitude: \[ |\vec{a}| = 3 \] Unit vector: \[ \hat{a}=\frac{1}{3}(2,-1,2) \] Directional derivative: \[ \nabla\phi \cdot \hat{a} = -4 \]

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Problem 4: Taylor Expansion

Problem: Expand \( f(x,y)=e^x\cos y \) about (0,0) up to second order.

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\[ f(0,0)=1 \] \[ f_x(0,0)=1, \quad f_y(0,0)=0 \] \[ f_{xx}=1,\quad f_{yy}=-1,\quad f_{xy}=0 \] \[ f(x,y)=1+x+\frac{1}{2}(x^2-y^2) \]

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Problem 5: Jacobian

Problem: If \( x=r\cos\theta, y=r\sin\theta \), find Jacobian \( \frac{\partial(x,y)}{\partial(r,\theta)} \).

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\[ J= \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} \] \[ = r(\cos^2\theta+\sin^2\theta)=r \]

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Problem 6: Lagrange Multiplier

Problem: Find extrema of \( f=x^2+y^2 \) subject to \( x+y=1 \).

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\[ \nabla f=(2x,2y), \quad \nabla g=(1,1) \] \[ (2x,2y)=\lambda(1,1) \Rightarrow x=y \] Using constraint: \[ 2x=1 \Rightarrow x=y=\frac12 \] Minimum value: \[ f=\frac12 \]

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Problem 7: Implicit Differentiation

Problem: If \( x^2+y^2+xy=7 \), find \( \frac{dy}{dx} \).

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\[ 2x+2y\frac{dy}{dx}+x\frac{dy}{dx}+y=0 \] \[ (2y+x)\frac{dy}{dx}=-(2x+y) \] \[ \frac{dy}{dx}=-\frac{2x+y}{2y+x} \]

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Problem 8: Harmonic Function

Problem: Show \( u=\ln(x^2+y^2) \) is harmonic.

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\[ u_x=\frac{2x}{x^2+y^2} \] \[ u_{xx}=\frac{2(y^2-x^2)}{(x^2+y^2)^2} \] \[ u_{yy}=\frac{2(x^2-y^2)}{(x^2+y^2)^2} \] \[ u_{xx}+u_{yy}=0 \] Hence harmonic.

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Problem 9: Euler’s Theorem

Problem: Verify Euler’s theorem for \( u=(x^2+y^2)^{3/2} \).

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\[ u_x=3x(x^2+y^2)^{1/2} \] \[ u_y=3y(x^2+y^2)^{1/2} \] \[ xu_x+yu_y=3(x^2+y^2)^{3/2}=3u \]

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Problem 10: Maximum Directional Derivative

Problem: Find maximum rate of increase of \( f=xye^z \) at (1,1,0).

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\[ \nabla f= (ye^z)\mathbf{i} + (xe^z)\mathbf{j} + (xye^z)\mathbf{k} \] At (1,1,0): \[ \nabla f=(1,1,1) \] Maximum rate: \[ |\nabla f|=\sqrt3 \] Direction: along gradient vector.

Partial Differentiation: Part 2

Numerical Examples on Partial Differentiation

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Example 1: First Order Partial Derivatives

Problem: If \( z = 3x^2y + 2xy^3 - 5y \), find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).

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Differentiate w.r.t \(x\):

\[ \frac{\partial z}{\partial x} = 6xy + 2y^3 \]

Differentiate w.r.t \(y\):

\[ \frac{\partial z}{\partial y} = 3x^2 + 6xy^2 - 5 \]

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Example 2: Second Order Partial Derivatives

Problem: If \( z = x^3y^2 + 4xy \), find \( \frac{\partial^2 z}{\partial x^2} \).

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First derivative:

\[ \frac{\partial z}{\partial x} = 3x^2y^2 + 4y \]

Second derivative:

\[ \frac{\partial^2 z}{\partial x^2} = 6xy^2 \]

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Example 3: Mixed Partial Derivatives

Problem: If \( z = x^2y^3 \), verify mixed partial derivatives.

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\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ \frac{\partial^2 z}{\partial x \partial y} = 6xy^2 \]

Hence verified.

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Example 4: Chain Rule

Problem: If \( z = x^2 + y^2 \), where \( x = r\cos\theta \), \( y = r\sin\theta \), find \( \frac{\partial z}{\partial r} \).

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\[ z = r^2(\cos^2\theta + \sin^2\theta) = r^2 \] \[ \frac{\partial z}{\partial r} = 2r \]

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Example 5: Implicit Differentiation

Problem: If \( x^2 + y^2 + z^2 = 1 \), find \( \frac{\partial z}{\partial x} \).

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\[ 2x + 2z\frac{\partial z}{\partial x} = 0 \] \[ \frac{\partial z}{\partial x} = -\frac{x}{z} \]

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Example 6: Euler’s Theorem

Problem: Verify Euler’s theorem for \( z = x^2y^3 \).

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\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 5x^2y^3 = 5z \]

Hence verified.

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Example 7: Total Derivative

Problem: If \( z = x^2y + y^3 \), where \( x=t^2 \), \( y=t \), find \( \frac{dz}{dt} \).

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\[ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \] \[ = (2xy)(2t) + (x^2+3y^2) \] Substitute \(x=t^2, y=t\): \[ \frac{dz}{dt} = 5t^4 + 3t^2 \]

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Example 8: Tangent Plane

Problem: Find tangent plane to \( z=x^2+y^2 \) at (1,1,2).

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\[ f_x=2x,\quad f_y=2y \] \[ z-2=2(x-1)+2(y-1) \] \[ z=2x+2y-2 \]

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Example 9: Maxima and Minima

Problem: Find stationary points of \( z=x^2+y^2-4x-6y \).

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\[ 2x-4=0 \Rightarrow x=2 \] \[ 2y-6=0 \Rightarrow y=3 \]

Minimum at (2,3).

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Example 10: Jacobian

Problem: If \( u=x+y \), \( v=x-y \), find \( \frac{\partial(u,v)}{\partial(x,y)} \).

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\[ J= \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} =-2 \]